AIME 2023 II · 第 3 题

Solution 1

This solution refers to the Diagram section.

Let $\angle PAB = \angle PBC = \angle PCA = \theta,$ from which $\angle PAC = 90^\circ-\theta,$ and $\angle APC = 90^\circ.$

Moreover, we have $\angle PBA = \angle PCB = 45^\circ-\theta,$ as shown below:

Note that $\triangle PAB \sim \triangle PBC$ by AA Similarity. The ratio of similitude is $\frac{PA}{PB} = \frac{PB}{PC} = \frac{AB}{BC},$ so $\frac{10}{PB} = \frac{1}{\sqrt2}$ and thus $PB=10\sqrt2.$ Similarly, we can figure out that $PC=20$.

Finally, $AC=\sqrt{10^2+20^2}=10\sqrt{5}$, so the area of $\triangle ABC$ is

$$\frac12\cdot AB\cdot AC = \frac12\cdot (10\sqrt{5})^2 = \boxed{250}.$$ ~s214425

~MRENTHUSIASM

Solution 2

Since the triangle is a right isosceles triangle, $\angle B = \angle C = 45^\circ$.

Let the common angle be $\theta$. Note that $\angle PAC = 90^\circ-\theta$, thus $\angle APC = 90^\circ$. From there, we know that $AC = \frac{10}{\sin\theta}$.

Note that $\angle ABP = 45^\circ-\theta$, so from law of sines we have

$$\frac{10}{\sin\theta \cdot \frac{\sqrt{2}}{2}}=\frac{10}{\sin(45^\circ-\theta)}.$$ Dividing by $10$ and multiplying across yields

$$\sqrt{2}\sin(45^\circ-\theta)=\sin\theta.$$ From here use the sine subtraction formula, and solve for $\sin\theta$:

$$\begin{aligned} \cos\theta-\sin\theta&=\sin\theta \\ 2\sin\theta&=\cos\theta \\ 4\sin^2\theta&=\cos^2\theta \\ 4\sin^2\theta&=1-\sin^2\theta \\ 5\sin^2\theta&=1 \\ \sin\theta&=\frac{1}{\sqrt{5}}. \end{aligned}$$ Substitute this to find that $AC=10\sqrt{5}$, thus the area is $\frac{(10\sqrt{5})^2}{2}=\boxed{250}$.

~SAHANWIJETUNGA

Solution 3

Since the triangle is a right isosceles triangle, $\angle B = \angle C = 45^\circ$.

Do some angle chasing yielding:

• $\angle APB = \angle BPC = 135^\circ$
• $\angle APC=90^\circ$

We have $AC=\frac{10}{\sin\theta}$ since $\triangle APC$ is a right triangle. Since $\triangle ABC$ is a $45^\circ$-$45^\circ$-$90^\circ$ triangle, $AB=\frac{10}{\sin\theta}$, and $BC=\frac{10\sqrt{2}}{\sin\theta}$.

Note that $\triangle APB \sim \triangle BPC$ by a factor of $\sqrt{2}$. Thus, $BP = 10\sqrt{2}$, and $PC = 20$.

From Pythagorean theorem, $AC=10\sqrt{5}$ so the area of $\triangle ABC$ is $\frac{(10\sqrt{5})^2}{2}=\boxed{250}$.

~SAHANWIJETUNGA

Solution 4

Since the triangle is a right isosceles triangle, $\angle B = \angle C = 45^\circ$.

Notice that in triangle $PBC$, $\angle PBC + 45-\angle PCA = 45^\circ$, so $\angle BPC = 135^\circ$. Similar logic shows $\angle APC = 135^\circ$.

Now, we see that $\triangle APB \sim \triangle BPC$ with ratio $1:\sqrt{2}$ (as $\triangle ABC$ is a $45^\circ$-$45^\circ$-$90^\circ$ triangle). Hence, $\overline{PB}=10\sqrt{2}$. We use the Law of Cosines to find $AB$.

$$\begin{aligned} AB^2&=BP^2+AP^2-2ab\cos(APB) \\ &=100+200-2(10)(10\sqrt{2}\cos(135^\circ)) \\ &=300+200\cdot\sqrt{2}\cdot\frac{1}{\sqrt{2}} \\ &=500. \end{aligned}$$ Since $\triangle ABC$ is a right triangle, the area is $\frac{AB^2}{2}=\frac{500}{2}=\boxed{250}$.

~Kiran

Solution 5

Denote the area of $X$ by $[X].$ As in previous solutions, we see that $\angle APC = 90 ^\circ, \triangle BPC \sim \triangle APB$ with ratio $k = \sqrt{2}\implies$

$$\frac {PC}{PB} = \frac {PB}{PA} = k \implies PC = k^2 \cdot AP = 20 \implies [APC] = \frac {AP \cdot PC}{2} = 100.$$ $$[BPC] = k^2 [APB] = 2 [APB].$$ $$AB = BC, \angle PCA = \angle PAB \implies \frac {[APC]}{[APB]} = \frac {PC}{PA} = 2 \implies$$ $$[ABC] = [APB] + [APC] + [BPC] = [APC] \cdot (\frac {1}{2} + 1 + 2 \cdot \frac {1}{2}) = \frac {5}{2} \cdot [APC] = \boxed{250}.$$ vladimir.shelomovskii@gmail.com, vvsss

Solution 6

Denote $\angle PCA = \theta$. Then, by trig Ceva's:

$$\begin{aligned} \frac{\sin^3(\theta)}{\sin(90-\theta) \cdot \left(\sin(45-\theta)\right)^2} &= 1 \\ \sin^3(\theta) &= \cos(\theta) \cdot \left(\sin(45) \cos(\theta) - \cos(45) \sin(\theta)\right)^2 \\ 2\sin^3(\theta) &= \cos(\theta) \cdot \left(\cos(\theta) - \sin(\theta)\right)^2 \\ 2\sin^2(\theta) &= \cot(\theta) \cdot \left(1 - 2\sin(\theta)\cos(\theta)\right) \\ 2\sin^2(\theta) &= \cot(\theta) - 2\cos^2(\theta) \\ \cot(\theta) &= 2 \\ \sin(\theta) &= \frac{\sqrt{5}}{5}. \end{aligned}$$ Note that $\angle APC$ is a right angle. Therefore:

$$\begin{aligned} \sin(\theta) &= \frac{AP}{AC} \\ AC &= \frac{10}{\frac{\sqrt{5}}{5}} \\ &= 10\sqrt{5} \\ |ABC| &= \frac{AC^2}{2} \\ &= \boxed{250}. \end{aligned}$$ ~ConcaveTriangle

Solution 7

Notice that point $P$ is one of the two Brocard Points of $\triangle ABC$. (The angle equalities given in the problem are equivalent to the definition of a Brocard point.) By the Brocard point formula,

$$\begin{aligned} \cot(\phi) = \cot(A)+\cot(B)+\cot(C) \end{aligned}$$ , where $\phi$ is equal to $\angle PAB$.(This is also called the Brocard angle of triangle ABC). Because the triangle is an isosceles right triangle, the cotangents are easy to compute:

$$\begin{aligned}\cot(\phi) = 0 + 1 + 1 \\ \cot(\phi) = 2\end{aligned}$$ By definition, $\cot(\phi) = \frac{\cos(\phi)}{\sin(\phi)}$. By the Pythagorean identity, $\cos(\phi)=\frac{2\sqrt{5}}{5}$ and $\sin(\phi) = \cos(\phi)=\frac{\sqrt{5}}{5}$. Consider triangle $APB$. By the problem condition, $\angle PBA = 45-\phi$, so $\angle BPA = 135^{\circ}$

$$\begin{aligned}\sin{45-\theta} = \sin{45}\cos{\phi}-\cos{45}\sin{\phi} = \frac{\sqrt{10}}{10}\end{aligned}$$ Now, we can use the Law of Sines.

$$\begin{aligned} \frac{AP}{\sin{45-\theta}}&=\frac{AB}{\sin{135}} \\ 10 \sqrt{10} &= \sqrt{2} AB \\ AB &= 10 \sqrt{5} \end{aligned}$$ Therefore, the answer is

$$[ABC] = \frac{1}{2} (AB)^2 = \boxed{250}.$$ ~ewei12

Solution 8

Notice that $\angle APC = 90^{\circ}$, $\angle BPA = 135^{\circ}$, and $\angle CPB = 135^{\circ}$ (from Solution 4). Now let $a = 0$ and $p = 10i$. Then by the angle restrictions $c = m + 10i$ and $b = -n + (n+10)i$ for some $m, n$. Since $\angle BAC = 90^{\circ}$, $ci = b$, or $(m + 10i)i = -n + (n+10)i$. Therefore $n = 10$, $AB = \sqrt{10^2 + 20^2} = \sqrt{500}$, and $[ABC] = \frac{1}{2} AB^2 = \boxed{250}$.

~aliz

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=APSUN-9Z_AU

Video Solution 2 by Piboy

https://www.youtube.com/watch?v=-WUhMmdXCxU&t=26s&ab_channel=Piboy

Video Solution by The Power of Logic(#3 and #4)

https://youtu.be/dS9K1o4gCA0