Geometry: Euclidean MATH 3120, Spring 2017 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. All ﬁfteen prior axioms are used to prove neutral geometry resulsts. Axioms 16 through 22 are needed to fully prove all Euclidean results for plane geometry. Note that the axioms and theorems of a neutral geometry allow us to prove that, for an arbitrary line and arbitrary external point, there exists at least one parallell line through the point parallel to the line. Moreover, the alternate interior angles of parallel lines cut by a transversal are congruent which leads directly to the fact that two lines perpendicular to the same line are parallel. The problem isn’t having parallel lines in a neutral geometry. The problem is having too many.

1 Building Blocks of Euclidean Geometry

SMSG Parallel Postulate. Through a given external point there is at most one line parallel to a given line. When we assume the strict Euclidean parallel postulate, we move from the possibility of inﬁnitely many parallel lines to exactly one. This restriction immediately allows for the proof of several important theorems.

Theorem 1. (E) The Euclidean parallel postulate is equivalent to the converse of the alternate interior angle theorem. (Proved in neutral geometry section.)

Theorem 2. (E) If two parallel lines are cut by a transversal, then alternate interior angle are congruent. (Proved in neutral geometry section.)

Theorem 3. (E) The sum of the measures of the interior angles of triangle is 180◦.

Theorem 4.* The Euclidean parallel postulate is equivalent to Theorem 3: The angle sum for every triangle is 180◦. (Hint: for a neutral geometry, the angle sum for triangle was at most 180◦. What forces the equality in Euclidean space?)

Corollary to Theorem 4. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

Deﬁnition: Parallelogram. A quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel.

Theorem 5. (E) The opposite sides of a prallelgoram are congruent. (Hint: use triangle congruence.)

1 Theorem 6. (E)* If a transversal intersects three parallel lines in such a way as to make congruent segments between the paralllels, then every transversal interesecting these parallel lines will do likewise. (Hint: given three parallel lines l1, l2, and l3 and a transversal t1 that cuts intersects it, create an arbitray t2 that also intersects it. Then, create a third transversal parallel to t1 and use similar triangles.)

Corollary to Theorem 6. (E) If a transversal intersects n parallel lines (n > 2) in such a way as to make congruent segments between the paralllels, then every transversal interesecting these parallel lines will do likewise. (Hint: induction.)

Theorem 7: Median Concurrence Theorem.* The three medians of a triangle are concurent at a point called the centroid. (Hint: consider 4DFG with median DM. Note M is the midpoint of FG. Call the centroid point C. Then the measure of DC is exactly twice the measure of CM. This is true - but must be proven. Theorem 6 and its corollary along with this fact provide an outline for the proof.)

Corollary to the Median Concurrence Theorem. Medians of a triangle intersect at a point that is two-thirds the distance from any vertex to the midpoint of the opposite side.

2 Consequences of the Euclidean Parallel Postulate

The following theorems are direct consequences of the Euclidean parallel postulate, and their proofs are reasonably elementary. Two key concepts are equivalent to the Euclidean parallel postulate: that a rectangle exists, and that all triangles have angle sum of 180◦. The theorems below establish many common notions in Euclidean geometry.

Theorem 8. (E) Two lines parallel to another line are parallel to each other.

Theorem 9. (E) If a line intersects one of two parallel llines, then it interesects the other.

Theorem 10. Each diagonal of a parallelogram partitions the parallelogram into a pair of congruent triangles.

Theorem 11. The diagonals of a parallelogram bisect each other.

Theorem 12. The diagonals of a quadrilateral bisect each other if and only if the quadrilateral is a parallelogram. (Note that one direction has been proven in already Theorem 11.)

Theorem 13. A quadrilateral has a pair of opposite sides that are congruent and parallel if and only if the quadrilateral is a parallelogram. (Note that one direction is true by deﬁnition of a parallelogram.)

Deﬁnition. We call line segments parallel if they are contained in lines that are parallel.

2 Theorem 14. If a line segment has as its endpoints the midpoints of two sides of a triangle, then the segment is parallel to and one-half the length of the third side of the triangle.

Deﬁnition. A rhombus is a quadrilateral in which all four sides are congruent.

Theorem 15. The diagonals of a rhombus are perpendicular.

Theorem 16. If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus.

Deﬁnition. A median of a triangle is a line passing through a vertex and the midpoint of the side opposite that vertex.

Theorem 17. In a right triangle, the median from the right angle to the hypotenuse is one-half the length of the hypotenuse.

Theorem 18.* In a right triangle, one of the angles measures 30◦ if and only if the side opposite this angle is one-half the√ length of the hypotenuse. (Note: this result along with the Pythagorean Theorem gives us the 1−2− 3 ratio of side lengths for 30−60−90 right triangles. Recall that the Pythagorean Theorem is equivalent to the Euclidean parallel postulate.)

Theorem 19. The sum of the measures of the interior angles of a convex n-gon is (n − 2)(180◦).

Theorem 20. The sum of the measures of the exterior angles of a convex n-gon is 360◦.

Theorem 21.* The midpoints of the sides of a quadrilateral are the vertices of a convex parallelo- gram.

3 Area and Congruence

We have discussed several issues with Euclid. We have arrived at another: How do we deﬁne area? If rectangles have an area that is deﬁned as length times width, then we can calculate the area of triangles. Using triangles and rectangles, we can calcuate polygon areas. The existence of a rectangle is equivalent to Euclid’s parallel postulate, so our standard notion of area only exists in Euclidean geometry. In any other geometry, calculating an area will prove troublesome. However, we’re developing Euclidean geometry in this section. Please note that SMSG Axiom 20 states that “the area of a rectangle is the product of the length of its base and the length of its altitude.”

Theorem 22. (E) The area of a parallelogram is the product of the lengths of its base and height.

Theorem 23. The area of a right triangle is one-half the product of the lengths of its legs.

3 Theorem 24. (E) The area of a triangle is one-half the product of the any base and the correspond- ing height.

Deﬁnition: Trapezoid. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Note that there are two diﬀerent deﬁnitions of a trapezoid: posssessing exactly one pair of parallel sides vs. possessing at least one pair of parallel sides. The ﬁrst means no parallelograms are trapezoids while the second means all parallelograms are trapezoids. More common is the one deﬁned above where parallelograms are not trapezoids. High school teachers should note carefully which deﬁnition is used in their curriculum materials especially when using classrooms resources and activities from the interwebs.

Theorem 25. The area of a trapezoid is the product of its height and the arithmetic mean of its bases.

Theorem 26. The area of a rhombus is one-half the product of its diagonals.

Deﬁnition: Similar Polygons. Two polygons are similar provided (i) corresponding sides of each are in the same proportion, and (ii) corresponding interior angles are congruent. Note that requirement (i) implies a constant ratio of proportionality for any two pairs of side lengths.

Theorem 27: Basic Proportionality Theorem. (E)* A line parallel to one side of a triangle intersects the other two sides in two diﬀerent points if and only if it divides these sides into segments that are proportional. (The “two diﬀerent points” simply requires the point of intersection to not be the vertex.)

Theorem 28: AAA Triangle Similarity. (E) If the interior angles of one triangle are congruent to corresponding angles of a second triangle, then the triangles are similar.

Theorem 29: SAS Triangle Similarity. If an angle of one triangle is congruent to the correspond- ing angle of a second triangle, and the corresponding sides adjacent to these angles are in proportion, then the triangles are similar.

Theorem 30: SAS Triangle Similarity. If the lengths of the sides of one triangle are proportional to the corresponding side lengths of a second triangle, then the triangles are similar.

Theorem 31: Pythagorean Theorem. (E)* If a and b are the lengths of the legs of a right triangle the hypotenuse of which has length c, then a2 + b2 = c2.

4 5 Circles

Deﬁnitions. A circle is a set of points all of which equidistant from a given point which is called the center. The common distance between the circle and center point is called the radius of the circle.

Theorem 32. (E) In the Euclidean plane, three distinct points determine a line. (This proof requires both an existance proof and a uniqueness proof. Both are essential.)

Deﬁnition. A chord of a circle is a line segment joining two of the points of the circle. A secant of a circle is a line which contains a chord.

Deﬁnition. A tangent of a circle is a line which contains exactly one point of a circle.

Deﬁnition. A diameter of a circle is a chord that contains the center of the circle.

Theorem 33. If AB is a diameter and CD is any other chord of the same circle (which is not a diameter), then AB > CD.

Theorem 34. If a diameter of a circle is perpendicular to a chord of the circle, then the diameter bisects the chord.

Theorem 35. If a diameter of a circle bisects a chord of the circle (which is not a diameter), then the diameter bisects the chord.

Theorem 36. The perpendicular bisector of a chord of a circle contains a diameter of the circle.

Theorem 37. If a line is tangent to a circle, it is perpendicular to the line joining the point of tangency to the center of the circle.

Deﬁnition. Any angle whose vertex is the center of a circle is a central angle.

Deﬁnition. An arc is the set of points on a circle that lie between two points on the circle, including the points themselves. The degree measure of an arc is the measure of the central angle corresponding to the arc. Each two points on a circle create two arcs, a minor arc and a major arc, with the degree measure of the major arc greater than (or equal to) that of the minor arc.

_ _ Theorem 38: Arc Addition. (E) If ABC and CDE are arcs sharing only the same endpoint C, _ _ _ then m ABC +m CDE= m ACE.

Deﬁnition. An inscribed angle is the angle formed when two secant lines intersect on the circle.

5 Deﬁnition. An intercepted arc is the part of the circle that lies between two lines that intersect it. (Typically, the two lines contains rays of an angle.)

Theorem 39: Inscribed Angle Theorem. (E) The measure of an angle inscribed in an arc is one-half the measure of its intercepted arc. (Recall the measure of an arc is equal to the measure of the central angle corresponding to the arc.)

Corollary 1 to the Inscribed Angle Theorem. An angle inscribed in a semi-circle is a right angle.

Corollary 2 to the Inscribed Angle Theorem. Angles inscribed in the same or congruent arcs are congruent.

Theorem 40: Two-Chord Angle Theorem. (E) If two chords interesect in the interior of a circle to determine an angle, the measure of that angle is the average of the measures of the arcs intercepted by the angle and its vertical angles.

Theorem 41: Two-Secant Angle Theorem. If two secants interesect in the exterior of a circle to determine an angle, the measure of the angle at the point of intersection is one-half the positive diﬀerence of the two intercepted arcs.

←→ Theorem 42: Tangent-Chord Angle Theorem. If AB is tangent to a circle at A, and if AC is _ ◦ x◦ a chord of the circle with m AP C= x , then m∠BAC = 2 .

←→ ←→ Theorem 43: Tangent-Secant Angle Theorem. If AB is tangent to a circle at A, and if BD is a secant of the circle, then m∠ABD is half the positive diﬀerence of the two intercepted arcs.

Theorem 44: Two-Tangent Angle Theorem. The measure of an angle formed by two tangents drawn to a circle is one-half the positive diﬀerence of the measures of the intercepted arcs.

Corollary Two-Tangent Angle Theorem. (E) Tangent segments drawn to a circle from the same (exterior) point are congruent.

Theorem 45: Chord Segment Product Theorem. (E)* If two chords of a circle intersect, the products of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other.

Theorem 46: Secant Segment Product Theorem. If two secant lines are drawn to a circle from the same exterior point, then the product of the length of the secant segment and the lenth of its exterior portion is the same for both secants.

6 6 Triangles

Deﬁnition. The median of a triangle is the line through a vertex point and the midpoint of the opposite side.

Theorem 47: Median Concurrence. (E) The three medians of a triangle are concurrent. The concurrent point is called the centroid of the triangle. (The centroid is the “center of mass” of a triangle. If the triangle and its interior were made of metal with perfectly constant mass, thickness and density, the centroid would be the balance point for that object.)

Theorem 48: Perpendicular Bisector Concurrence. (E) The three perpendicular bisectors of the sides a triangle are concurrent. The concurrent point is called the circumcenter of the triangle.

Corally to Perpendicular Bisector Concurrence.* The circumcircle of a triangle passes through all three vertices of the triangle. It’s center is the circumcenter. (Note the circumcircle circumscribes the triangle.)

Deﬁnition. The distance from a point P to a line l is the distance from P to the foot of the of perpendicular drawn from P to l. We often instruct students to think intuitively about the distance from a point to a line as the “minimum distance” from P to l. This idea is equivalent to the deﬁtion because any other line segment drawn from P to l would be the hypotenuse of a right triangle. The Pythagorean Theorem then implies the new segment has a length greater than the distance from P to l.

Theorem 49.* A point is on the bisector of angle if and only if it is equidistant from both rays of the angle.

Theorem 50. A point is on the bisector of angle if and only if it is equidistant from both rays of the angle.

Theorem 51: Angle Bisector Concurrence. (E)* The three bisectors of the interior angles of a triangle are concurrent. The concurrent point is called the incenter. (Hint: make use of Theorem 50.)

Theorem 51: Angle Bisector Concurrence.* The incircle of a triangle is the unique circle which can be inscribed in that triangle (e.g. tangent to all three sides of the triangle and otherwise interior to the triangle).

Deﬁnition. An altitude of a triangle is the perpendicular line sement from the vertex of an angle to the side opposite.

Theorem 52: Altitude Concurrence. (E)* The three altitudes of a triangle are concurrent. The concurrent point is called the orthocenter.

7 Theorem 53. (E) An angle bisector of an interior angle of a triangle is concurrent with the angle bisectors of the exterior angles of the two remaining angles of the triangles. This is true for all three interior angles. The point of concurrence is the same for all three interior angles, and the concurrent point is called the excenter.

Corollary to Theorem 53. Each excenter of a triangle is the center of a circle which is externally tangent to one side of the triangle and tangent to the extensions of the other two sides.

Theorem 54. (E)* In an equilateral triangle, the incenter, circumcenter, centroid and orthocenter are all concurrent.

Theorem 55: Euler Line.* The orthocenter, centroid and circucenter of a triangle are collinear. The line that contains these three points is called the Euler Line.

7 The Geometric Mean

Deﬁnition: Geometric Mean. For two real numbers a and b, their geometric is the unique real number x such that: a x = x b

Theorem. The middle term in three consecutive terms of a geometric series is the geometric mean of the other two.

8 Homework Problems

Deﬁnitions • A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel. – The parallel sides of a trapezoid are called the bases. – The non-parallel sides of a trapezoid are called the legs. – Base angles of a trapezoid refer to a pair of angles whose vertices are the endpoints of the same base segment. – The median of a trapezoid is the line segment joining the midpoints of the legs. • An isosceles trapezoid has two pairs of congruent base angles. • A kite is a quadrilateral with two distinct pairs of adjacent, congruent sides. • A cyclic quadrilateral is a quadrilateral the vertices of which lie on a single circle. 1. The median of a trapezoid is parallel to the bases and its length is half the sum of the lengths of the bases. 2. A trapezoid is isosceles if and only if its legs are congruent.

8 3. A trapezoid has congruent diagonals if and only if it is isosceles. 4. The line segment joining the midpoints of the bases of an isosceles trapezoid is perpendicular to both. 5. Opposite angles of a trapezoid are supplementary if and only if it is isosceles. 6. A quadrilateral is a convex kite if and only if one of its diagonals is the perpendicular bisector of the other. 7. In a convex kite, the diagonals are perpendicular. 8. In a convex kite, there exists a pair of opposite angles congruent. 9. In every kite, there exists a diagonal that - along with the sides of the kite - forms two isosceles triangles. 10. A convex kite has at least one diagonal that bisects a pair of opposite angles. 11. A quadrilateral is a parallelogram if and only if two pairs of opposite sides are congruent. 12. Conjecture: A quadrilateral is a parallelogram if and only if every pair of adjecent angles is supplementary. Prove or disprove. 13. A quadrilateral is a parallelogram if and only if its diagonals bisect each other. 14. A quadrilateral is a parallelogram if and only if there exists a pair of opposite sides that are both parallel and congruent. 15. A quadrilateral is a rectangle if and only if its diagonals are congruent. (Deﬁnition: a “rectangle” is a parallelogram with a right angle.) 16. A rectangle has four right angles. 17. A quadrilateral is a rhombus if and only if its diagonals are perpendicular. (Deﬁnition: a “rhom- bus” is a parallelogram with all sides congruent.) 18. A quadrilateral is a rhombus if and only if both its diagonals bisect a pair of opposite angles. 19. A quadrilateral is a square if and only if it is a rectangle that is also a rhombus. (Deﬁnition: A “square” is a rectangle with congruent sides.) 20. A quadrilateral is a square if and only if it is a rhombus with a right angle. 21. A quadrilateral is a square if and only if it is a rhombus with two adjacent sides perpendicular. 22. Create a conjecture about the diagonals of a square, and prove it. 23. Opposite angles of a cyclic quadrilateral are supplementary. 24. If opposite angles of a qularilateral are supplementary, then the quadrilateral is cyclic. 25. The exterior angle of a cyclic quadrilateral is congruent to the opposite interior angle. 26. The diagonals of cyclic quadrilateral ABCD meet at X. Prove 4ADX ∼ 4BCX. −−→ −→ −→ 27. Rays AB and AC are tangent to a circle at B and C. Chord BD is drawn from B k AC. Prove that 4BCD is isosceles. 28. The point A is in the interior of a circle with center O and radius r. Prove that CA × AD = r2 − (OA)2 where DC is any chord through A.

9 29. The diagonals of trapezoid ABCD with bases AB k CD meet at the point P . The legs of the trapezoid are tangent to the circle through points D,P,C at points D and C. Prove: ∼ (a) ∠BAC = ∠ADB. (b) AB is tangent to the circle through points D,P,A. (c) Quadrilateral ABCD is cyclic. 30. Prove or disprove. A trapezoid is a cyclic quadrilateral if and only if it is isoceles. ←→ 31. 4ABC is inscribed in a circle with RS tangent to the circle at point C with R − C − S. Prove: ∼ (a) ∠RCA = ∠B, and ∼ (b) ∠SCB = ∠A. 32. The circumscribing circle of a cyclic quadrilateral can be constructed by ﬁnding the circumcenter of the quadrilateral. Prove that the circumcenter of the cyclic quadrilateral is the circumcenter of any of the triangles formed by using three of the four vertices from the quadrilateral. 33. Law of Sines. In any 4ABC, a b c = = = 2r sin A sin B sin C a where r is the radius of the circumcircle of 4ABC. (Hint: prove sin A = 2r using three cases.) 34. All four sides of quadrilateral ABCD are tangent to an inscribed circle. Prove that opposite sides of ABCD are congruent. 35. A common chord of two overlapping circles joins the points of intersection of the circles and is thus a chord for both. Prove that the three common chords of three mutually overlapping circles are concurrent. Note that we don’t neccessarily have three common chords in general, so the circles must intersect in ways that produce three common chords.

10