What is the formal proof of AA similarity test?

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It is a well-known fact that if any two triangles are equiangular then their sides are proportional and converse. But I am not sure how to prove it by using plane geometry only without using trigonometry, vectors, etc. Please help.

I want to use only the tools of Euclidean geometry.

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1 Answer

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Given : Triangles ABC and DEF with $\angle A=\angle D$, $\angle B=\angle E$ and $\angle C=\angle F$

Step 1:

Construction:

Take the points P and Q on AB and AC such that $AP=DE$ and $AQ=DF$

Step 2:

$AP=DE$

$\angle A=\angle D$

$AQ=DF$

$\triangle APQ\cong \triangle DEF$ (SAS)

Hence $\angle P=\angle E$ and $\angle Q=\angle F$

Step 3:

$PQ\parallel BC$ (converse, corresponding angles, $\angle P=\angle B$ and $\angle Q=\angle C$)

Step 4:

$\frac {AP}{AB}= \frac {AQ}{AC}$ (Converse, Proportionally Theorem, $PQ\parallel BC$)

and

$\frac {DE}{AB}= \frac {DF}{AC}$ (AP=DE and AQ=DF)

Step 5:

Similirly by constructing points on the other sides of triangle ABC

$\frac{DE}{AB}=\frac{DF}{AC}=\frac{EF}{BC}$

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