## Sunday, July 19, 2009

### Problem 325. Isosceles triangle, Altitude, Incircle, Excircle

Proposed Problem
Click the figure below to see the complete problem 325 about Isosceles triangle, Altitude, Cevian, Incircle, Excircle with equal radius. See more:
Complete Problem 325
Level: High School, SAT Prep, College geometry

1. let be M midpoint of AC, K point BC meet circle F

draw MT tangent to circle F =>
T, F, K are at the same line (FK perpend to BC, & FT to MT) => TK diameter ( MT//BK)

extend TM to G on AH => MG middle line of tr ACH
=> G midpoint of AH =>

GH = AH = TK = 2r => AH = 4r

2. Why "T, F, K are at the same line (FK perpend to BC, & FT to MT)"???

3. from a point , F, we can draw one ( only ) perpendicular to two parallel lines ( GT//HK )

4. draw ST tg to F and // to AC
name G, K, L, E meet AD, AB, BD
name M, N, R, P, F meet DC, DS, ST, CT
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▲AEG=▲AEK=▲RFT=▲PFT => AK=AG=RT=PT ( anlge & radius )
▲GED=▲DEL=▲DMF=▲DNF => GD=DM=DL=DN
MC = CP tg from C, NS = SR tg from S
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BN = BP => BD + DM = BC + MC =>
BD + DM = AB + MC (1)
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SADB + SBST = SABC + SDCTS