I sincerely request the experts to answer this:

Prove that in an equilateral triangle, all the altitudes are equal.

Please don't answer as:

Prove that if all the altitudes in a triangle are equal, then it is an equilateral triangle.

Question

I sincerely request the experts to answer this:
​

Prove that in an equilateral triangle, all the altitudes are
equal

Please don't answer as:

Prove that if all the altitudes in a triangle are equal, then it is
an equilateral triangle

Shreha Arora · Accepted Answer

Dear Student,

Let ABC be the triangle

where AD, BE and CF are the three altitudes on sides BC, AC and AB
respectively

In triangle BEC and triangle CFB

angle BEC = angle CFB (both 90o)

angle C = angle B (angles of equilateral triangle are equal)

BC = CB (common)

so, triangle BEC congruent triangle CFB (by AAS congruency)

or BE = CF (by CPCT) -----(1)

similarly, it can be shown that triangle ABE congruent triangle BAD
by AAS

and BE = AD (by CPCT) -----(2)

from (1) and (2)

AD = BE = CF

Regards

I sincerely request the experts to answer this: ​ Prove that in an equilateral triangle, all the altitudes are equal. Please don't answer as: Prove that if all the altitudes in a triangle are equal, then it is an equilateral triangle.