MHB Convex Quadrilateral Problem: Prove Inequality with Diagonal Point T

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In the convex quadrilateral problem, the goal is to prove that for quadrilateral ABCD with equal sides AB and AD, and a point T on diagonal AC where the angles ABT and ADT sum to angle BCD, the inequality AT + AC ≥ AB + AD holds. Anemone will be grading this week's problem and will resume posting next week. The thread also acknowledges MarkFL for temporarily taking over POTW duties. Additionally, it notes that last week's problem went unanswered, with a suggested solution provided below. The discussion emphasizes the importance of engaging with the problem and following the guidelines for participation.
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Here is this week's POTW:


Let $ABCD$ be a convex quadrilateral with $\overline{AB}=\overline{AD}$. Let $T$ be a point on the diagonal $\overline{AC}$ such that $\angle ABT+\angle ADT=\angle BCD$.

Prove that $\overline{AT}+\overline{AC}\ge\overline{AB}+\overline{AD}$.


anemone will be grading this week's problem, and will be back to posting the problems next week. (Smile)

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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I would like to give a vote of thanks to MarkFL for standing in for me while I was unavailable to handle my POTW duties. (Handshake) (Smile)

No one answered last week's problem. You can find the suggested solution below:
View attachment 7881
On the segment $\overline{AC}$, consider the unique point $T'$ such that $\overline{AT}'\cdot\overline{AC}=\overline{AB}^2$. The triangles $ABC$ and $AT'B$ are similar: they have the angle at $A$ common and $\overline{AT}':\overline{AB}=\overline{AB}:\overline{AC}$. So $\angle ABT'=\angle ACB$. Analogously, $\angle ADT'=\angle ACD$. So $\angle ABT'+\angle ADT'=\angle BCD$. But $ABT'+ADT'$ increases strictly monotonously, as $T'$ moves from $A$ towards $C$ on $\overline{AC}$. The assumption on $T$ implies that $T'=T$. So, by the arithmetic-geometric mean inequality,$$\overline{AB}+\overline{AD}=2\overline{AB}=2\sqrt{\overline{AT}\cdot\overline{AC}}\le\overline{AT}+\overline{AC}$$
 

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