SELECTION and TRAINING SESSION

Question

The medians AM and BN of a triangle ABC are the diameters of the circles _1 and _2. If _1 touches the altitude CH prove that _2 also touches CH.

Accepted Answer

The statement easily follows from the following fact: if points

A_{1}

,

B_{1}

are chosen on the sides

A C

,

B C

of the triangle

A B C

, respectively, then the radical axis of the circles with the diameters

A A_{1}

and

B B_{1}

contains the orthocenter of the triangle

A B C

.

In case when

A A_{1}

and

B B_{1}

are medians the centerline of

ω_{1}

and

ω_{2}

is obviously parallel to the side

A B

, then the radical axis is perpendicular to

A B

, thus the statement follows.