Problem B. 5294. (February 2023)

B. 5294. Two altitudes of an acute-angled triangle \(\displaystyle ABC\) are \(\displaystyle AT_A\) and \(\displaystyle BT_B\). The midpoint of \(\displaystyle AB\) is \(\displaystyle F\), and the midpoint of \(\displaystyle T_AT_B\) is \(\displaystyle G\). Prove that \(\displaystyle FG\) is perpendicular to \(\displaystyle T_AT_B\).

Proposed by V. Vígh, Sándorfalva

(3 pont)

Deadline expired on March 10, 2023.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Tekintsük az \(\displaystyle AB\) szakasz Thalész-körét: ennek középpontja \(\displaystyle F\) és áthalad a \(\displaystyle T_A\) és \(\displaystyle T_B\) pontokon (lásd az ábrát).

Ennek a körnek tehát \(\displaystyle T_AT_B\) egy húrja, a húr felezőpontját (\(\displaystyle G\)) a kör középpontjával összekötő szakasz (\(\displaystyle FG\)) pedig mindig merőleges a húrra.

Statistics:

123 students sent a solution.
3 points:	93 students.
2 points:	12 students.
1 point:	3 students.
0 point:	12 students.

Problems in Mathematics of KöMaL, February 2023