Quadrilateral Or Triangle?

Geometry Level 3

The figure above shows an incircle of a triangle A B C ABC with sides 6-12-17. Let P P and Q Q be the points on the side lengths A B AB and A C AC respectively, such that P Q PQ is tangent to the incircle. Find the perimeter of the triangle A P Q APQ .


The answer is 11.

Dragan Marković by Dragan Marković , 26, Serbia

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2 solutions

Ahmad Saad
Apr 27, 2016

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Very nice solution! (+1)

Dragan Marković - 5 years, 1 month ago
Dragan Marković
Apr 26, 2016

P B C Q PBCQ is a tangent quadrilateral so P B + Q C = P Q + B C ( 1 ) PB+QC=PQ+BC \qquad (1) . Wanted perimeter is O = P Q + A P + A Q O=PQ+AP+AQ . Next P B = A B − A P PB=AB-AP and Q C = A C − A Q QC=AC-AQ . If we put that in ( 1 ) (1) we get:

A B − A P + A C − A Q = P Q + B C ⟹ A B + A C − B C = P Q + A P + A Q ⟹ 17 + 6 − 12 = O AB-AP+AC-AQ=PQ+BC\implies AB+AC-BC=PQ+AP+AQ \implies 17+6-12=O Thus O = 11 O=11 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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