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Regional Olympiad - FBH 2016 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

September 22, 2018

geometrycollinearsemicircletouching circles

Problem Statement

Let

k

be a diameter of semicircle

h

. On this semicircle there is point

C

, distinct from points

A

and

B

. Foot of perpendicular from point

C

to side

AB

is point

D

. Circle

k

is outside the triangle

ADC

and at the same time touches semicircle

h

and sides

AB

and

CD

. Touching point of

k

with side

AB

is point

T

, with semicircle

b)

is point

T

and with side

CD

is point

S

a)

Prove that points

A

,

S

and

T

are collinear

b)

Prove that

AC=AE

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