O is any point outside the equilateral triangle ABC and within the angular region on ABC; OP, OQ and OR are the perpendicular distance of AB, BC and CA respectively from the point O. Let us prove that the altitude of the triangle = OP + OQ – OR.

Given.

OP, OQ and OR are the perpendicular distance of AB, BC and CA respectively from the point O

Formula used.

Area of triangle = × Base × Height

As triangle ABC, triangle OAB, triangle OBC combines to form triangle OAC

Area of triangle ABC = Area of triangle OCB + triangle OAB – triangle OAC

Area of triangle ABC = × Base × Height

× AC × BD

Area of triangle OAC = × Base × Height

× AC × OR

Area of triangle OAB = × Base × Height

× AB × OP

Area of triangle OCB = × Base × Height

× BC × OQ

⇒ × AC × BD = ×BC×OQ + ×AB×OP – × AC × OR

As AB = AC = BC (Given)

⇒ × AC × BD = × BC×OQ + ×AC×OP – ×AC×OR

Taking common ×AC get removed

∴ BD = OQ + OP – OR

Hence proved;