MathDB

4

Part of 2021 Mexico National Olympiad

Problems(1)

two circles tangent at the euler line

Source: Mexico National Olympiad 2021Problem 4

11/10/2021
Let ABCABC be an acutangle scalene triangle with ∠BAC=60∘\angle BAC = 60^{\circ} and orthocenter HH. Let ωb\omega_b be the circumference passing through HH and tangent to ABAB at BB, and ωc\omega_c the circumference passing through HH and tangent to ACAC at CC.
[*] Prove that ωb\omega_b and ωc\omega_c only have HH as common point. [*] Prove that the line passing through HH and the circumcenter OO of triangle ABCABC is a common tangent to ωb\omega_b and ωc\omega_c.
Note: The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.
geometrytangent circlesorthocenterEuler Linecircumcircle