A strange connection in triangles?

Casually I found a relation between the sides of any triangle but I don't know if it is a thing that you can obtain from other theorems or no

Given a triangle $ABC$ I draw the height $BD$ and I call $W$ the value $W=\frac{AB^2+BC^2-AC^2}{2}$ (that is equal to $AB*BC*\cos(\hat{B})$ because of the law of cosines)

Then you can find (I've found this casually, but you can verify easily thanks to GeoGebra or similar programs) $BC^2=DC*AC+W$ and on the other side $AB^2=AD*AC+W$ but I can't explain myself these equalities, could you help me please? Thanks :)

#Geometry

Easy Math Editor

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Comments

Check out stewart's theorem.

Calvin Lin Staff - 5 years ago

Let's put $AB=c \\ BC=a \\ CA=b$

Then the conjectured equations are equivalent to the following equations: $a=b\cos C + c\cos B \\ c=a\cos B + b\cos A$

But these equivalent equations are true for any arbitrary triangles (try proving this).

$\large Q. E. D.$

A Former Brilliant Member - 5 years ago

Just as a hint, try drawing an altitude from one of the vertices to a side, and then use the two right triangles formed on either side.

Ameya Daigavane - 5 years ago

Yes I already know these equations, but I don't understand how they can be equivalent to the ones I wrote (I tried to put them equal but I can't reach a result)

Matteo Monzali - 5 years ago

Notice that $AD=c \cos A$ and $CD=a \cos C$ .

A Former Brilliant Member - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$