MathDB

In a triangle

ABC

, let

M_{a}

M_{b}

M_{c}

be the midpoints of the sides

BC

CA

AB

, respectively, and

T_{a}

T_{b}

T_{c}

be the midpoints of the arcs

BC

CA

AB

of the circumcircle of

ABC

, not containing the vertices

A

B

C

, respectively. For

i \in \left\{a, b, c\right\}

, let

w_{i}

be the circle with

M_{i}T_{i}

as diameter. Let

p_{i}

be the common external common tangent to the circles

w_{j}

and

w_{k}

(for all

\left\{i, j, k\right\}= \left\{a, b, c\right\}

) such that

w_{i}

lies on the opposite side of

p_{i}

than

w_{j}

and

w_{k}

do. Prove that the lines

p_{a}

p_{b}

p_{c}

form a triangle similar to

ABC

and find the ratio of similitude.
Proposed by Tomas Jurik, Slovakia

Problem Statement