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Two circles are tangents in a triangles with angle 60
nAalniaOMliO   1
N Mar 29, 2025 by sunken rock
Source: Belarusian National Olympiad 2025
In a triangle $ABC$ angle $\angle BAC = 60^{\circ}$. Point $M$ is the midpoint of $BC$, and $D$ is the foot of altitude from point $A$. Points $T$ and $P$ are marked such that $TBC$ is equilateral, and $\angle BPD=\angle DPC = 30^{\circ}$ and this points lie in the same half-plane with respect to $BC$, not in the same as $A$.
Prove that the circumcircles of $ADP$ and $AMT$ are tangent.
Ivan Korshunau
1 reply
nAalniaOMliO
Mar 28, 2025
sunken rock
Mar 29, 2025
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Two circles are tangents in a triangles with angle 60
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Source: Belarusian National Olympiad 2025
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nAalniaOMliO
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nAalniaOMliO
#1 h  Mar 28, 2025, 8:27 PM
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In a triangle $ABC$ angle $\angle BAC = 60^{\circ}$. Point $M$ is the midpoint of $BC$, and $D$ is the foot of altitude from point $A$. Points $T$ and $P$ are marked such that $TBC$ is equilateral, and $\angle BPD=\angle DPC = 30^{\circ}$ and this points lie in the same half-plane with respect to $BC$, not in the same as $A$.
Prove that the circumcircles of $ADP$ and $AMT$ are tangent.
Ivan Korshunau
This post has been edited 2 times. Last edited by nAalniaOMliO, Apr 4, 2025, 4:47 PM
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sunken rock
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sunken rock
#2 h  Mar 29, 2025, 2:33 PM
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Observation only: $T$ is circumcenter of $\triangle BPD$, I do not know if it will be helpful.
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