Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle

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Theorem

In the words of Euclid:

If there be three plane angles of which two, taken together in any manner, are greater than the remaining one, and they are contained by equal straight lines, it is possible to construct a triangle out of the straight lines joining the extremities of the equal straight lines.

(The Elements: Book $\text{XI}$: Proposition $22$)


Proof

Euclid-XI-22a.png

Let $\angle ABC, \angle DEF, \angle GHK$ be plane angles such that the sum of any two is greater than the remaining one.

That is:

$\angle ABC + \angle DEF > \angle GHK$
$\angle DEF + \angle GHK > \angle ABC$
$\angle GHK + \angle ABC > \angle DEF$

Let the straight lines $AB, BC, DE, EF, GH, GK$ be equal.

Let $AC$, $DF$ and $GK$ be joined.

It is to be demonstrated that it is possible to construct a triangle from straight lines equal to $AC$, $DF$ and $HK$.


If $\angle ABC, \angle DEF, \angle GHK$ are equal to one another, then it is possible to construct an equilateral triangle from $AC$, $DF$ and $HK$.

Euclid-XI-22b.png

Otherwise, let $\angle ABC, \angle DEF, \angle GHK$ be unequal.

On the straight line $HK$ at the point $H$, let:

$\angle KHL$ be constructed equal to $\angle ABC$
$HL$ be constructed equal to one of $AB, BC, DE, EF, GH, GK$
$KL$ be joined.

We have that:

$AB$ and $BC$ are equal to $KH$ and $HL$
$\angle ABC = \angle KHL$

Therefore from Proposition $4$ of Book $\text{I} $: Triangle Side-Angle-Side Congruence:

$AC = KL$

We have that:

$\angle GHK + \angle ABC > \angle DEF$

while:

$\angle ABC = \angle KHL$

Therefore:

$\angle GHL > \angle DEF$

We have that:

$GH$ and $HL$ are equal to $DE$ and $EF$
$\angle GHL > \angle DEF$

Therefore from Proposition $24$ of Book $\text{I} $: Hinge Theorem:

$GL > DF$

But:

$GK + KL > GL$

Therefore:

$GK + KL > DF$

But:

$KL = AC$

Therefore:

$AC + GK > DF$

Similarly it can be proved that:

$AC + DF > GK$

and that:

$DF + GK > AC$

Hence the result.

$\blacksquare$


Historical Note

This proof is Proposition $22$ of Book $\text{XI}$ of Euclid's The Elements.


Sources

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