MHB Geometry Challenge: Prove $\angle ADE=\angle BDC$ in Convex Quadrilateral $ADBE$

AI Thread Summary
In convex quadrilateral $ADBE$, a point $C$ is located within triangle $ABE$ such that the angles satisfy $\angle EAD + \angle CAB = \angle EBD + \angle CBA = 180^{\circ}$. This configuration leads to the conclusion that $\angle ADE = \angle BDC$. The discussion highlights the importance of geometric properties and relationships in proving angle congruence. Additionally, participants share tips on using TiKZ for effective diagram creation, emphasizing the simplicity of basic commands for drawing geometric figures. Overall, the thread combines mathematical proof with practical advice on diagramming techniques.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
In convex quadrilateral $ADBE$, there is a point $C$ within $\triangle ABE$ such that $\angle EAD+\angle CAB=\angle EBD+\angle CBA=180^{\circ}$.

Prove that $\angle ADE=\angle BDC$.
 
Mathematics news on Phys.org
[TIKZ]
\begin{scope}
\draw (0,0) circle(3);
\end{scope}
\coordinate[label=left: E] (E) at (-3,0);
\coordinate[label=below: D] (D) at (1.2,-2.75);
\coordinate[label=below: A] (A) at (-1,.-2.828);
\coordinate[label=right: F] (F) at (2.9,-0.768);
\coordinate[label=above: B] (B) at (-1,-0.26);
\coordinate[label=above: C] (C) at (-2,-1.1);
\draw (A) -- (E);
\draw (A) -- (D);
\draw (F) -- (D);
\draw (E) -- (F);
\draw (A) -- (B);
\draw (A) -- (F);
\draw (E) -- (D);
\draw (B) -- (D);
\draw [dashed] (C) -- (D);
\draw [dashed] (C) -- (B);
\draw [dashed] (C) -- (A);
[/TIKZ]

Let F be the second intersection of the circumcircle of $\triangle EAD$ and line $EB$. Then $\angle DBF=180^{\circ}-\angle EBD=\angle CBA$. Moreover,

$\begin{align*}\angle BDF&=180^{\circ}-\angle AEB-\angle ADB\\&=180^{\circ}-(360^{\circ}-\angle EAD-\angle EBD)\\&= 180^{\circ}-(\angle CAB+\angle CBA)\\&=\angle BCA\end{align*}$

These two relations give $\angle BDF \simeq \triangle BCA$.

So $\dfrac{BD}{BF}=\dfrac{BC}{BA}$ Together with $\angle DBF=\angle CBA$, we have $\triangle BDC \simeq \triangle BFA$.

This results in $\angle ADE=\angle AFE=\angle BFA=\angle BDC$. (Q.E.D.)
 
You always do such a nice job with your presentations...the TiKZ drawings are really nice (and add such quality), and I know they take some effort too. :)
 
Mark, to be completely honest, I have to say once you get to know some simple commands like how to draw a circle, joining lines, labeling angles, coloring some region, etc, then basically you can draw anything out of these simple commands. Of course, my other trick is always look for Klaas for help when I got stuck in some effect I want to produce to my diagram, hehehe... (Happy)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top