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 1 of 4: Use the pedal-angle lemma
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In plain words

Use the pedal-angle lemma

∠APB−∠C=∠XZY\angle APB-\angle C=\angle XZY
Detailed analysis

Let X,Y,Z X,Y,Z be the perpendicular feet from P P to BC,CA,AB BC,CA,AB , respectively, and write A=∠BAC A=\angle BAC , B=∠ABC B=\angle ABC , C=∠ACB C=\angle ACB . Cyclic quadrilaterals AYPZ AYPZ and the right angles at X,Y,Z X,Y,Z give ∠APB−C=∠XZY\angle APB-C=\angle XZY and ∠APC−B=∠XYZ\angle APC-B=\angle XYZ .