Tell whether the following statement is true or false:

AD, BE, CF are the altitudes of triangle ABC such that AD = BE = CF, then triangle ABC is an equilateral triangle.

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True

False

answer is A.

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Detailed Solution

Given AD, BE, CF are the altitudes of triangle ABC such that AD = BE = CF.
For the triangles BEC and BFC,
BE = CF (given)

∠ B E C = ∠ B F C

(given)
BC = BC (common in both triangles)
Therefore, triangles BEC and BFC are congruent triangles using RHS postulate.
Now, as triangles are congruent, so

∠ A B C = ∠ A C B

.
Now, for the triangles ADB and ADC,

∠ A D B = ∠ A D C

(given)
AD = AD (common in both triangles)

∠ A B C = ∠ A C B

(proved above)
Therefore, triangles ADB and ADC are congruent triangles using ASA postulate.
Now, as triangles are congruent, so AB = AC. Similarly, triangles BFC and CFA are congruent triangles using ASA postulate.
Now, as triangles are congruent, so BC = AC.
Therefore, AB = AC = BC.
Hence, ABC is an equilateral triangle.
Given statement is true.
Hence, option 1 is correct.

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