PROOF 57

Prove.

The converse of the Pythagorean theorem is also true

On the Euclidean plane.

A way to read this proof

First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.

What you may use

A1q→p does not follow from p→q
the converse is not equivalent to the original statement (Proof 25)
A2∠C = 90° ⟹ a² + b² = c²
Pythagorean theorem (Proof 56)
A3three sides equal ⟹ congruent
SSS congruence condition
A4a triangle exists from two sides and the included angle
constructibility — given two sides and the included angle
A5sum of two sides > the third ⟹ exists
condition for a triangle with three given sides
A6congruent ⟹ corresponding angles equal
meaning of congruence
A7side lengths of a triangle > 0
positivity of length
A8x,y>0, x²=y² ⟹ x=y
positive numbers correspond one-to-one with their squares
Ea = b ⟹ same operation on both sides
properties of equality

D1the converse of p→q is q→p
definition of the converse
D2equilateral and isosceles triangles
three sides equal / two sides equal
T1the converse does not follow
Theorem 1 — a counterexample
T2a²+b²=c² ⟹ ∠C = 90°
Theorem 2 — the converse of Pythagoras

Nothing outside this list may be cited.

Definition D1

For a statement p → q, the statement q → p is called its converse.

Proof.

(1)The converse exchanges hypothesis and conclusion; it refers to the form only.convention

(2)The truth of the original statement does not determine the truth or falsity of its converse.(1), A1

∎

Definition D2

A triangle with all three sides equal is called equilateral; a triangle with two sides equal is called isosceles.

Proof.

(1)An equilateral triangle is a special case of an isosceles triangle.convention

∎

Theorem T1

"If equilateral, then isosceles" is true, but its converse is false. Hence a converse does not follow from the original statement.

Proof.

(1)Let p be "T is equilateral" and q be "T is isosceles".convention, D2

(2)If all three sides are equal, then some two of them are equal, so p → q is true.(1), D2

(3)2 + 2 > 3 and 2 + 3 > 2, so a triangle T₀ with sides 2, 2, 3 exists.A5

(4)T₀ has two equal sides, so it is isosceles, and its three sides are not all equal, so it is not equilateral.(3), D2

(5)For T₀, q is true and p is false, so q → p is false.(1), (4), D1

(6)p → q is true while q → p is false. Hence the converse does not follow from the original statement.(2), (5), A1

(7)Therefore the truth of A2 alone does not yield "if a² + b² = c², then ∠C = 90°". The converse must be proved separately.(6), A1, A2, D1

∎

Theorem T2

In triangle ABC, let a = BC, b = CA, c = AB. If a² + b² = c², then ∠C = 90°.

Proof.

(1)Suppose a² + b² = c².assumption

(2)There exists a triangle A′B′C′ with C′B′ = a, C′A′ = b, ∠C′ = 90°.A4

(3)Triangle A′B′C′ has a right angle at C′, so A′B′² = a² + b²(2), A2

(4)A′B′² = c²(1), (3), E

(5)A′B′ > 0 and c > 0(2), A7

(6)A′B′ = c(4), (5), A8

(7)BC = C′B′ = a, CA = C′A′ = b, AB = A′B′ = c, so the two triangles have three respectively equal sides.(2), (6), convention

(8)Triangles ABC and A′B′C′ are congruent.(7), A3

(9)Corresponding angles are equal, so ∠C = ∠C′ = 90°(2), (8), A6

∎

Remark

Can A3 be removed from this proof?

This proof draws a new triangle and sets it against the original. Check what is needed to set two triangles against each other.