MA 341 – Topics in Geometry Lecture 22 Theorems 1. A convex quadrilateral is cyclic if and only if the four perpendicular bisectors of the sides are concurrent. 2. A convex quadildrila tera l is cyclic if and only if opposite angles are suppltlementary.
21-Oct-2011 MA 341 001 2 Ptolemy’s Theorem Let a, b, c, and d be the b lhlengths of consecutive sides of a cyclic y a quadrilateral and let x x c and y be the lengths of the diagonals. Then ac + bd = xy. d Also, xad+bc = y ab +cd
21-Oct-2011 MA 341 001 3 Ptolemy’s Theorem This leads to the b fllfollowing: y (()()ac +bd)(ad+bc) a x= x c ab +cd and d (ac +bd)(ab + cd) y = adbd +bc
21-Oct-2011 MA 341 001 4 Angles of a cyclic quadrilateral If A is the angle between a and d a+d-b-c2222 cos(A) = 2(ad +bc) (s-a )(s- b)(s-c )(s- d) sin(A) =2 ad+bc A(s-a)(s-d) tan = 2(s2 (s- b)(s- c)
21-Oct-2011 MA 341 001 5 Angles of a cyclic quadrilateral The angle, θ, between the diagonals
(s -b)(s -d) tan = 2(s-a)(s-c)
The circumradius is 1 (ab +cd)(ac +bd)(ad+bc) R= 4 (s - a)(s -b)(s -c)(s -d) (Ph’Parameshvara’s fl)formula)
21-Oct-2011 MA 341 001 6 Incenters of the triangles
The incenters of 4 triangles form a rectangle
Sides paralle l to lines through midarc poitints
21-Oct-2011 MA 341 001 7 Incenters of the triangles
The incenters and excenters of 4 triangles form a 4 x 4 grid
21-Oct-2011 MA 341 001 8 Incenters of the triangles
Consider quadrilaterals formed by centroids, 9-point circle centers and orthocenters of the 4 triangles
21-Oct-2011 MA 341 001 9 Incenters of the triangles
All are similar to ABCD.
H2H1H4H3 is congruent to ABCD.
21-Oct-2011 MA 341 001 10 Tangential Quadrilaterals
A convex quadrilateral whose sides all lie tangent to inscribed circle – the incircle
21-Oct-2011 MA 341 001 11 Tangential Quadrilaterals
Is a rectangle a tangential quadrilateral?
Is a square a tangential quadrilateral?
Is a parall el ogram a tangential quadrilateral?
21-Oct-2011 MA 341 001 12 Tangential Quadrilaterals
Is a rhombus a tangential quadrilateral?
Is a kite a tangential quadrilateral?
21-Oct-2011 MA 341 001 13 Tangential Quadrilaterals
In triangle incenter = intersection of angle bisectors
What about quadrilaterals?
21-Oct-2011 MA 341 001 14 Tangential Quadrilaterals I is equidistant from AD, AB and BC B N C
∆API ∆AMI PI MI M
∆MIB ∆NIB NI MI D P A
21-Oct-2011 MA 341 001 15 Tangential Quadrilaterals I is equidistant from AD, AB, BC and CD B N C
∆API ∆AMI PI MI M
∆MIB ∆NIB NI MI etc D Thus, there is an inscribed P circle! A
21-Oct-2011 MA 341 001 16 Tangential Quadrilaterals
If there is an inscribed circle, then center is equidistant from sides.
21-Oct-2011 MA 341 001 17 Tangential Quadrilaterals
Then, BI, CI, DI, and AI are angle bisectors.
21-Oct-2011 MA 341 001 18 Theorem
A quadrilateral is N tangential if and only if a + c = b + d. M O
P
21-Oct-2011 MA 341 001 19 Proof
BM=BN N CN=CO DO= DP M AP=AM O a + c = AM+MB+CO+OD = AP+BN+CN+DP = b + d P
21-Oct-2011 MA 341 001 20 Proof B Assume a + c = b + d. B’ a A
CsttConstruct ciilrcle tangent to AB, AD, CD. b d Construct CB’ tangent to circle WLOG AB’ 21-Oct-2011 MA 341 001 23 Area For inradius r and N semiperimeter s K = rs M O P 21-Oct-2011 MA 341 001 24 Area Trigonometric formula p K = rs BBD+D KbdK =abcd sin 2 q 1 K= pq22 -(ac-bd) 2 2 21-Oct-2011 MA 341 001 25 Further Results 1. If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, the line passes through the incenter 2. If M and N are midpoints of diagonals and incenter = I, then M, I, and N are collinear 21-Oct-2011 MA 341 001 26 Further Results 3. If incircle is tangent to AB, BC, CD, DA at X, Y, Z, W respectively, then lines XY, WZ and AC are concurrent 4. If I is incenter then IA IC+ IB ID= AB BC CD DA 5. Incenter I of ABCD coincides with centroid of ABCD if and only if IA IC = IB ID 21-Oct-2011 MA 341 001 27 Further Results Look at the incircles of the triangles of a tangential quadrilateral. 21-Oct-2011 MA 341 001 28 Further Results 21-Oct-2011 MA 341 001 29