Question

If the altitude drawn from the vertices of abc to the opposite sides are equal, prove that the triangle is equilateral.

Answer 1

If the altitude drawn from the vertices of abc to the opposite sides are equal, prove that the triangle is equilateral.

• Consider the attached figure,
• Given,
• ABC is a triangle
• A, B, C are the vertices of triangle
• AD, BE, CF are the altitudes of triangle
• Consider, Δ BEC and Δ BFC,
• BC = BC (common sides)
• BE = CF (given)
• ∠B = ∠C (angles of equal sides are equal)
• ∴ Δ BEC ≅ Δ BFC (SAS theorem)
• ⇒ AC = AB ......(1) (sides opposite to equal angles are equal)
• Similarly,
• Δ BEC ≅ Δ BFC
• ∠B = ∠A
• AC = BC  .........(2)
• From (1) and (2) we have,
• AB = BC = AC
• Therefore Δ ABC is an equilateral triangle.