Problems

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

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Let ABC and DEF be two triangles such that ACB=DFE and |DF||AC|=|EF||BC|. Prove that ABC and DEF are similar.

Let AA1 and BB1 be the medians of the triangle ABC. Prove that A1B1C and BAC are similar. What is the similarity coefficient?

Let AD and BE be the heights of the triangle ABC, which intersect at the point F. Prove that AFE and BFD are similar.

Let AD and BE be the heights of the triangle ABC. Prove that DEC and 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?

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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 ABC with right angle ACB=90, 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.