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 3 of 4: Convert to a side ratio
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In plain words

Convert to a side ratio

ABPB=ACPC\dfrac{AB}{PB}=\dfrac{AC}{PC}
Detailed analysis

The sine rule in ABC ABC gives ABsin⁡B=ACsin⁡C AB\sin B=AC\sin C . Combining this with PCsin⁡C=PBsin⁡B PC\sin C=PB\sin B yields AB/PB=AC/PC AB/PB=AC/PC .