# In an isosceles triangle ABC (AC-base), point O is the midpoint of side BC.

**In an isosceles triangle ABC (AC-base), point O is the midpoint of side BC. On side AC, point D is marked such that 2DO = BC. Find AC if CD = 1**

Since point O is the middle of the segment BC, then BO = CO, and then 2 * OC = BC.

According to the condition, the length of the segment two lengths of DO are equal to BC, 2 * DO = BC, then 2 * OC = 2 * DO, and therefore OC = DO, and the triangle DOC is isosceles.

Since the triangle DОС is isosceles, the angles at its base DC are equal, and since the triangle ABC is isosceles, the angle BAC = ОDC = ОD.

Then the triangles ABC and DOC are similar in two angles, and the coefficient of their similarity is: K = BC / OC = 2/1.

Then AC / CD = 2.

AC = 2 * CD = 2 * 1 = 2 cm.

Answer: The length of the base of the speaker is 2 cm.