MathDB

3

Part of 2023 South East Mathematical Olympiad

Problems(2)

Prove collinear

Source: 2023 China South-east MO Grade 10 P3

8/1/2023
In acute triangle ABCABC (△ABC\triangle ABC is not an isosceles triangle), II is its incentre, and circle ω \omega is its inscribed circle. ⊙ω\odot\omega touches BC,CA,ABBC, CA, AB at D,E,FD, E, F respectively. ADAD intersects with ⊙ω\odot\omega at JJ (J≠DJ\neq D), and the circumcircle of △BCJ\triangle BCJ intersects ⊙ω\odot\omega at KK (K≠JK\neq J). The circumcircle of △BFK\triangle BFK and △CEK\triangle CEK meet at LL (L≠KL\neq K). Let MM be the midpoint of the major arc BACBAC. Prove that M,I,LM, I, L are collinear.
geometryincenter
Beautiful conditional geo in China Southeast

Source: 2023 China Southeast Grade 11 P3

7/30/2023
In △ABC\triangle {ABC}, D{D} is on the internal angle bisector of ∠BAC\angle BAC and ∠ADB=∠ACD\angle ADB=\angle ACD. E,FE, F is on the external angle bisector of ∠BAC\angle BAC, such that AE=BEAE=BE and AF=CFAF=CF. The circumcircles of △ACE\triangle ACE and △ABF\triangle ABF intersects at A{A} and K{K} and A′A' is the reflection of A{A} with respect to BCBC. Prove that: if AD=BCAD=BC, then the circumcenter of △AKA′\triangle AKA' is on line ADAD.
geometryangle bisector