In $\triangle{ABC}$, $AM$ is the median on the side $BC$. Prove that $AB^2+AC^2=2(AM^2 + BM^2)$
For $\triangle{ABC}$, $O$ is an inner point, and $D$, $E$, $F$ are on $BC$, $CA$, $AB$ respectively, such that $OD\perp BC$, $OE\perp CA$, and $OF\perp AB$. Prove that $AF^2+BD^2+CE^2=BF^2+DC^2+AE^2$.
$P$ is an interior point of $\triangle{ABC}$, $P_1$, $P_2$, and $P_3$ are exterior points outside of $AB$, $BC$, and $CA$, respectively. $PP_1\perp AB$, $PP_2\perp BC$, $PP_3 \perp AC$, and $BP_1 = BP_2$, $CP_2=CP_3$. Prove that $AP_1=AP_3$.
In square $ABCD$, $M$ is the midpoint of $AD$ and $N$ is the midpoint of $MD$. Prove that $\angle{NBC}=2\angle{ABM}$.
In $\triangle{ABC}$, $\angle{A}=90^\circ$, $AB=AC$, $D$ is a point on $BC$. Prove that $BD^2+CD^2 = 2AD^2$.
In $\triangle{ABC}$, $\angle{C}=90^\circ$, $D$ is the midpoint of $AC$. Prove that $AB^2+3BC^2=4BD^2$.
In $\triangle{ABC}$, $\angle{C}=90^\circ$, $E$, $D$ are points on $AC$ and $BC$ respectively. Prove that $AD^2+BE^2=AB^2+DE^2$.
In $\triangle{ABC}$, $\angle{C}=90^\circ$, $D$ is the midpoint of $AB$, $E$, $F $are two points on $AC$ and $BC$ respectively, and $DE\perp DF$. Prove that $EF^2=AE^2+BF^2$.
Let $ABCD$ be a convex quadrilateral. Prove that $AC\perp BD$ if and only if $AB^2+CD^2=AD^2+BC^2$.