Calculate the perimeter of triangle ABC.

Analysis:

• The image shows a triangle ABC with an inscribed circle.
• The points where the circle touches the sides are labeled E, M, and F.
• The lengths of the tangent segments from each vertex are given:
• AE = 4, AF = 4
• BE = 10, BF = 10
• CM = 6, CE = 6

The sides of the triangle are formed by the sums of these tangent segments:

• Side AB = AE + EB = 4 + 10 = 14
• Side BC = BF + FC = 10 + 6 = 16 (Note: The diagram has BF and CM as tangent lengths from B and C to the circle on side BC. This appears inconsistent with the other labels. Let's assume BF refers to the tangent segment from B to the circle on side BC, and CM refers to the tangent segment from C to the circle on side BC. However, the diagram labels points E, M, F as tangency points. Let's use the vertex to tangency point lengths as indicated.)

Re-evaluating based on standard notation for tangent segments from vertices to an inscribed circle:

• From vertex A, tangent segments to sides AB and AC are AE and AF. So, AE = AF = 4.
• From vertex B, tangent segments to sides BA and BC are BE and BF. So, BE = BF = 10.
• From vertex C, tangent segments to sides CA and CB are CF and CM. So, CF = CM = 6.

Now, let's calculate the side lengths:

• Side AB = AE + EB = 4 + 10 = 14
• Side BC = BF + FC = 10 + 6 = 16
• Side AC = AF + FC = 4 + 6 = 10

The perimeter of triangle ABC is the sum of its side lengths:

Perimeter = AB + BC + AC = 14 + 16 + 10 = 40.

Ответ: 40