## Tuesday, January 25, 2011

### Problem 573: Cyclic quadrilateral, Angle bisector, Perpendicular

Geometry Problem
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1. name K arc BC meet EH, L arc CD meet FH
in ▲ EFH
H = C - ( HED + HFB )= C - ( CD/2 - CK/2 + MB/2 - CL/2 )
H = C - ( C/2 - A/2 ) = C/2 + A/2
H = 90

2. a correction
CD/2 is GD/2 ( third row )

3. let EG intersect BC at K and AD at L respectively.
angle HLF = angle A + angle BEL (by exterior angle property)
angle HKF = angle KCE + angle KEC.
but angle KEC = angle BEL since EL bisects angle BEC.
and angle KCE = angleA because in a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
these imply that angle HLF = angle HKF.
also HF is common to the triangles HLF and HKF.
and angle HFL = angle HFK because FM bisects angle AFB.
so by AAS criterion the triangles HLF and HKF are congruent. so it implies that angle KHF = angle LHF. but they constitute a linear pair so angle LHF = angle KHF = 90 degrees.
therefore EG is perpendicular to FM.
Q. E. D.

4. Let G meet BC and AD at X and Y respectively. Then < X and < Y are both equal to <A + E/2 since external < C = A and so Tr. FXY is isoceles and the result follows

Sumith Peiris
Moratuwa
Sri Lanka