Question 208755
In Triangle ABC, AD is a median.
Prove that AB^2 + AC^2 = 2AD^2 + 2DC^2.
<pre><font size = 4 color = "indigo"><b>
{{{drawing(400,460,-5,5,-5,5,

triangle(-4,-3,4,4,0,4), locate(-4,-3,B),locate(0,1/2,D),
locate(4,4,C),locate(0,4.4,A),

triangle(0,1/2,4,4,0,4)


 )}}}

{{{AB^2=BD^2+AD^2}}} by the Pythagorean theorem

{{{AC^2=DC^2+AD^2}}} by the Pythagorean theorem

{{{AB^2+AC^2=BD^2+DC^2 + 2AD^2}}} because equals 
                 added to equals 
                 give equals.

{{{BD=DC}}} because Median AD bisects side BC

{{{AB^2+AC^2=DC^2+DC^2 + 2AD^2}}}  Substituting DC for BD

{{{AB^2+AC^2=2DC^2+ 2AD^2}}}   Combining like terms.

Edwin</pre>