Chapter 5 the Incircle and Excircles

Chapter 5

The incircle and excircles

5.1 The incircle

The internal angle bisectors of a triangle are concurrent at the incenter of the triangle. This is the center of the incircle, the circle tangent to the three sides of the triangle. Let the bisectors of angles B and C intersect at I. Consider the pedals of I on the three sides. Since I is on the bisector of angle B, IX = IZ. Since I is also on the bisector of angle C, IX = IY . It follows IX = IY = IZ, and the circle, center I, constructed through X, also passes through Y and Z, and is tangent to the three sides of the triangle.

A A

s a − s a − Y Y

Z I Z I s c − s b −

B X C B s b X s c C − − This is called the incircle of triangle ABC, and I the incenter. Let s be the semiperimeter of triangle ABC. The incircle of triangle ABC touches its sides BC, CA, AB at X, Y , Z such that

AY =AZ = s a, − BZ =BX = s b, − CX =CY = s c. − The inradius of triangle ABC is the radius of its incircle. It is given by 2∆ ∆ r = = . a + b + c s 150 The incircle and excircles

Example. If triangle ABC has a right angle at C, then the inradius r = s c. −

s b −

I r s a r −

s c r −

C s c s a A − − It follows that if d is the diameter of the incircle, then a + b = c + d.

Exercise 1. A square of side a is partitioned into 4 congruent right triangles and a small square, all with equal inradii r. Calculate r.

2. Calculate the radius of the congruent circles in terms of the sides of the right triangle. 5.1 The incircle 151

3. The incenter of a right triangle is equidistant from the midpoint of the hypotenuse and the vertex of the right angle. Show that the triangle contains a 30◦ angle. C

B A

4. The circle BIC intersects the sides AC, AB at E and F respectively. Show that EF is tangent to the incircle of triangle ABC. 1

Z I F

B C X

5. The median BE of triangle ABC is trisected by its incircle. Calculate a : b : c.

E I

B C

6. ABC is an isosceles triangle with a : b : c =4:3:3. Show that its orthocenter lies on the incircle.

B C

1Hint: Show that IF bisects angle AF E. 152 The incircle and excircles

7. The triangle is isosceles and the three small circles have equal radii. Suppose the large circle has radius R. Find the radius of the small circles. 2

OA TX ZT 8. The three small circles are congruent. Show that each of the ratios AB , XY , TO . is equal to the golden ratio.

Z X Y

T B O A

9. The large circle has radius R. The four small circles have equal radii. Calculate this common radius.

2 2 2R sin θ cos θ Let θ be the semi-vertical angle of the isosceles triangle. The inradius of the triangle is 1+sin θ = 2R sin θ(1 sin θ). If this is equal to R (1 sin θ), then sin θ = 1 . From this, the inradius is 3 R. − 2 − 4 8 5.2 The excircles 153

5.2 The excircles

The internal bisector of each angle and the external bisectors of the remaining two angles are concurrent at an excenter of the triangle. An excircle can be constructed with this as center, tangent to the lines containing the three sides of the triangle.

B C

ra Z ra

Ia The exradii of a triangle with sides a, b, c are given by ∆ ∆ ∆ r = , r = , r = . a s a b s b c s c − − − 1 1 1 The areas of the triangles IaBC, IaCA, and IaAB are 2 ara, 2 bra, and 2 cra respectively. Since ∆= ∆IaBC +∆IaCA +∆IaAB, − we have 1 ∆= ra( a + b + c)= ra(s a), 2 − − ∆ from which ra = s a . − 154 The incircle and excircles

5.3 Heron’s formula for the area of a triangle

Consider a triangle ABC with area ∆. Denote by r the inradius, and ra the radius of the excircle on the side BC of triangle ABC. It is convenient to introduce the semiperimeter 1 s = 2 (a + b + c).

ra r

C A Y ′ Y

(1) From the similarity of triangles AIY and AI′Y ′, r s a = − . ra s

(2) From the similarity of triangles CIY and I′CY ′,

r ra =(s b)(s c). · − − (3) From these,

(s a)(s b)(s c) r = − − − , r s s(s b)(s c) r = − − . a s a r − Theorem 5.1 (Heron’s formula).

∆= s(s a)(s b)(s c). − − − Proof. ∆= rs. p Proposition 5.1.

α (s b)(s c) α s(s a) α (s b)(s c) tan = − − , cos = − , sin = − − . 2 s(s a) 2 bc 2 bc s − r r 5.3 Heron’s formula for the area of a triangle 155

Exercise 1. If the incenter is equidistant from the three excenters, show that the triangle is equi- lateral.

2. The altitudes a triangle are 12, 15 and 20. What is the area of the triangle ? 3

3. Find the inradius and the exradii of the (13,14,15) triangle.

4. If one of the ex-radii of a triangle is equal to its semiperimeter, then the triangle contains a right angle.

5. Let ABC be a triangle with A = 108◦ and B = C = 36◦. Show that AIa : BIb : CIc = a : b : c. 6. Show that the line joining vertex A to the point of tangency of BC with the A-excircle intersects the incircle at the antipode of its point of tangency with BC.

C B Aa

Ia 7. ABC is an isosceles triangle with AB = AC and a : b : c =2: ϕ : ϕ. Show that the circumcircle and the A-excircle are orthogonal to each other, and ﬁnd the ratio ra : R.

O B C

3triangle = 150. The lengths of the sides are 25, 20 and 15. 156 The incircle and excircles

8. The length of each side of the square is 6a, and the radius of each of the top and bottom circles is a. Calculate the radii of the other two circles. 4

9. Show that in a right triangle the twelve points of contact of the inscribed and escribed circles form two groups of six points situated on two circles which cut each other orthogonally at the points of intersection of the cirucmcircle with the line joining the midpoints of the legs of the triangle.

C A

4 3 a and 4 a. 5.3 Heron’s formula for the area of a triangle 157

10. ABCD is a square of unit side. P is a point on BC so that the incircle of triangle ABP and the circle tangent to the lines AP , PC and CD have equal radii. Show that the length of BP satisﬁes the equation

2x3 2x2 +2x 1=0. − −

D C

A B

11. ABCD is a square of unit side. Q is a point on BC so that the incircle of triangle ABQ and the circle tangent to AQ, QC, CD touch each other at a point on AQ. Show that the radii x and y of the circles satisfy the equations

x(3 6x +2x2) y = − , √x + √y =1. 1 2x2 − Deduce that x is the root of

4x3 12x2 +8x 1=0. − −

D C

A B