MathLabs

Problem 2

Let P P be a point inside triangle ABC ABC such that ∠APB−∠ACB=∠APC−∠ABC\angle APB-\angle ACB=\angle APC-\angle ABC . Let D D and E E be the incenters of triangles APB APB and APC APC , respectively. Show that AP AP , BD BD , and CE CE meet at a point.
Step 2 of 4: Exploit the hypothesis
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In plain words

Exploit the hypothesis

XY=XZXY=XZ
Detailed analysis

The hypothesis makes the two angles in the last display equal, so triangle XYZ XYZ is isosceles and XY=XZ XY=XZ . Applying the sine rule in the right triangles around the pedal triangle gives PCsin⁡C=PBsin⁡B PC\sin C=PB\sin B .