Problems

In a triangle $\mathrm{△}ABC$, the angle $\mathrm{\angle }B={90}^{\circ }$ . The altitude from point $B$ intersects $AC$ at $D$. We know the lengths $|AD|=9$ and $|CD|=25$. What is the length $|BD|$?

Let $\mathrm{△}ABC$ and $\mathrm{△}DEF$ be two triangles such that $\mathrm{\angle }ACB=\mathrm{\angle }DFE$ and $\frac{|DF|}{|AC|}=\frac{|EF|}{|BC|}$. Prove that $\mathrm{△}ABC$ and $\mathrm{△}DEF$ are similar.

Let $A{A}_1$ and $B{B}_1$ be the medians of the triangle $\mathrm{△}ABC$. Prove that $\mathrm{△}{A}_1{B}_{1}C$ and $\mathrm{△}BAC$ are similar. What is the similarity coefficient?

Let $AD$ and $BE$ be the heights of the triangle $\mathrm{△}ABC$, which intersect at the point $F$. Prove that $\mathrm{△}AFE$ and $\mathrm{△}BFD$ are similar.

Let $AD$ and $BE$ be the heights of the triangle $\mathrm{△}ABC$. Prove that $\mathrm{△}DEC$ and $\mathrm{△}ABC$ are similar.

The area of the red triangle is $25$ and the area of the orange triangle is $49$. What is the area of the trapezium $ABCD$?

Prove that the ratio of perimeters of similar polygons is equal to the similarity coefficient.

Let $CB$ and $CD$ be tangents to the circle with the centre $A$, let $E$ be the point of intersection of the line $AC$ with the circle. Draw $FG$ as the segment of a tangent drawn through the point $E$ between the lines $CB$ and $CD$. Find the length $|FG|$ if the radius of the circle is $15$ and $|AC|=39$.

Prove that the relation between areas of two similar polygons equals to the square of their similarity coefficient.

In triangle $\mathrm{△}ABC$ with right angle $\mathrm{\angle }ACB={90}^{\circ }$, $CD$ is the height and $CE$ is the bisector. Draw the bisectors $DF$ and $DG$ of the triangles $BDC$ and $ADC$. Prove that $CFEG$ is a square.