AC^2 Algebraic simplifications

Question

Suppose I know that:

$AB^2=DF^2+CF^2$; $AD^2=BE^2+CE^2$; $AC^2=AE^2+CE^2$; $AC^2=AF^2+CF^2$

And also: $DF=AF-AD$ and $BE=AE-AB$

How can I prove that $AB * AE + AD * AF = AC^2$?

I start like this, but then I get stucked:

$AB*AE + AD*AF = AB*(BE+AB) + AD*(DF+AD) = AB*BE + AB^2 + AD*DF + AD^2 = AB*BE + AB^2 + AD*DF + BE^2 + CE^2 = (AE - BE)*BE + AB^2 + AD*DF + BE^2 + CE^2 = AE*BE - BE^2 + AB^2 + AD*DF + BE^2 + CE^2 = AE*BE + AB^2 + AD*DF + CE^2 = AE*(AE - AB) + AB^2 + AD*DF + CE^2 = AE^2 - AE*AB + AB^2 + AD*DF + CE^2...$

Can anyone give a hint how to continue?

Thanks!

Answer 1

Let $AB=x$, $BE=y$, $AD=p$, $DF=q$, now from the first $4$ equations we have \begin{eqnarray*} p^2+(x+y)^2=y^2+CE^2+(x+y)^2=y^2+AC^2 \\=y^2+(p+q)^2+CF^2=y^2+p^2+2pq+x^2 \end{eqnarray*} and so $xy=pq$.

Now \begin{eqnarray*} AB \times AE+ AD \times AF = x(x+y)+p(p+q) \end{eqnarray*} and \begin{eqnarray*} AC^2 = (x+y)^2+CE^2=(x+y)^2+p^2-y^2 \end{eqnarray*} and thus $\color{red}{AB \times AE+ AD \times AF =AC^2}$.

Answer 2

The situation described in the question is of a parallelogram $ABCD$ with obtuse angles at $B,D$. Construct a circle with diameter $AC$. Extend side $AB$ from $A$ to the circle intersecting at $E.\;$ Extend side $AD\;$ fom $A$ to the circle intersecting at $F.\;$ The segment $AF$ is a chord of the circle. Draw another chord $CG$ from $C$ parallel to segment $AE.\;$So $\;CD\!=\!AB, DG\!=\!BE, AD\!=\!BC$.

The Intersecting chords theorem gives $AD\cdot DF=CD\cdot DG.\;$ Now $AD\cdot DF=AB\cdot BE.\;$ Let $\;t:=\cos(\pi-\angle B).\;$ By the law of cosines, $\;AC^2 = AB^2 + BC^2 + 2AB\cdot BC\cdot t.\;$ Also, by using trigonometry of cosine, $\;t\cdot BC=BE\;$ and $\;t\cdot CD=DF.\;$ Use algebra to calculate $$AB\!\cdot\!AE\!+\!AD\!\cdot\! AF\!= \!AB\!\cdot\!(AB\!+\!BE)\!+\!BC\!\cdot\!(BC\!+\!DF)\!= \!AB^2\!+\!BC^2\!+\!AB\!\cdot\!BE\!+\!BC\!\cdot\!DF.$$ We are now done since $\;AB \cdot BC \cdot t=AB\cdot BE=BC\cdot DF.$ This Euclidean geometry result is somewhat similar to Ptolemy's theorem about sum of products of sides of a cyclic quadrilateral.