PROOF 150

Prove.

The triangle area formula comes from the rectangle

On the Euclidean plane.

What you may use

A1area of a rectangle = width × height
the standard of area
A2congruent figures have equal area
area of congruent figures
A3non-overlapping pieces: areas add
additivity of area
A4every polygon has an area
existence of area
A5a rectangle on the two legs of a right angle
existence of the rectangle (in the line of Euclid I.46)
A6opposite sides of a rectangle are equal
Euclid I.34
A7three equal sides ⟹ congruent
SSS congruence condition (Euclid I.8)
A8a diagonal splits a quadrilateral in two
diagonal decomposition of a convex quadrilateral — the two triangles do not overlap and fill the quadrilateral
A9joining a point on the opposite side splits in two
decomposition of a triangle — the segment from a vertex to a point on the opposite side makes two non-overlapping triangles filling the original
A10the foot of the perpendicular exists uniquely
perpendicular from a point off a line
A11the perpendicular makes right angles with points of ℓ
perpendicularity — the angle at the foot with any other point of the line is right
A12Q between ⟺ PQ+QR = PR
addition of segments
A13one of three points is between
order axiom — of three distinct points on a line, exactly one lies between the other two
A14associativity, commutativity, distributivity
arithmetic of the reals
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

Call one side BC of △ABC the base and write b for its length. Let H be the foot of the perpendicular from A to line BC, call the length of AH the height, and write h for it.

Proof.

Related labs

Perimeter & Area lab →