Proved that

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April 20th 2010, 09:13 AM
dapore

Proved that

http://www.saifoali.org/up/files/rvp...fi4qdvs2ss.bmp
April 20th 2010, 01:23 PM
Soroban

Hello, dapore!

Quote:

In any triangle $ABC$ with angles $A,B,C$ and sides $a,b,c\!:$

. . . $b\cos A + a\cos B \:=\:c$

Code:

C o *| * * | * * | * b * | * a * | * * | * * | * A o * o * * * o B x D y : - - - - - c - - - - - :

Construct altitude $CD$ to side $AB.$
Let $x = AD,\;Y = DB$

In right triangle $CDA\!:\;\;\cos A \,=\,\frac{x}{b} \quad\Rightarrow\quad x \,=\,b\cos A$ .[1]

In right triangle $CDB\!:\;\;\cos B \,=\,\frac{y}{a} \quad\Rightarrow\quad y \,=\,a\cos B$ .[2]

Note that: . $x + y \:=\:c$ .[3]

Substitute [1] and [2] into [3]: . $b\cos A + a\cos B \;=\;c$
April 20th 2010, 02:38 PM
dapore

Quote:

Originally Posted by Soroban

Hello, dapore!

Code:

C o *| * * | * * | * b * | * a * | * * | * * | * A o * o * * * o B x D y : - - - - - c - - - - - :

Construct altitude $CD$ to side $AB.$
Let $x = AD,\;Y = DB$

In right triangle $CDA\!:\;\;\cos A \,=\,\frac{x}{b} \quad\Rightarrow\quad x \,=\,b\cos A$ .[1]

In right triangle $CDB\!:\;\;\cos B \,=\,\frac{y}{a} \quad\Rightarrow\quad y \,=\,a\cos B$ .[2]

Note that: . $x + y \:=\:c$ .[3]

Substitute [1] and [2] into [3]: . $b\cos A + a\cos B \;=\;c$

Thanks Soroban
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