Proposed Problem

Click the figure below to see the complete problem 325 about Isosceles triangle, Altitude, Cevian, Incircle, Excircle with equal radius.

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Complete Problem 325

Level: High School, SAT Prep, College geometry

## Sunday, July 19, 2009

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

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

ReplyDeletedraw 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

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

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

ReplyDeletedraw ST tg to F and // to AC

ReplyDeletename 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

(AB+AD+BD)∙r+(BS+ST+BT)∙r=BC∙h+(DC+CT+ST+DS)∙r

(AB+AD+BD+BS+BT)∙r=BC∙h+(DC+CT+BT)∙r

(AB+AD+BD+BD+DM+NS+BC+CT)∙r=BC∙h+(DC+CT+BT)∙r

(3∙AB+AD+BD+MC+NS)∙r=BC∙h+(DC+DS)∙r

(3∙AB+AG+DM+BK+DN+MC+NS)∙r=BC∙h+(DC+DS)∙r

(4∙AB+DC+DS)∙r=BC∙h+(DC+DS)∙r

4∙AB∙r = BC∙h

4∙r = h

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