Problem 63. In an isosceles trapezoid $ABCD$, a circle is inscribed, touching the sides of the trapezoid at points $T_{1}, T_{2}, T_{3}, T_{4}$ (see Fig.53). $K$ is the point of intersection of $CT_{3}$ and $T_{1}T_{2}$. In what ratio does point $K$ divide $T_{1}T_{2}$?
Solution. By Archimedes' Lemma ( $T_{3} T_{4}$ is the diameter, $T_{2} E$ is the perpendicular dropped from the point of tangency $T_{2}$ ), the line $C T_{3}$ bisects $T_{2} E$. Therefore, $T_{1} K$ is to $K T_{2}$ as $3: 1$.

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