For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$

Question

In the interior of $\triangle ABC$ we take the arbitrary point $P$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\angle BCP + \sin\angle BAP)\\ \le a + b + c$$

I tried to draw the distances from $P$ to the sides of the triangle and write the sines from the right angled triangles, but I couldn't get anything out of it. I saw that a case of equality is when the triangle is equilateral and $P$ is its center.

Can you help me?

Answer 1

To simplify derivation introduce the notation: $$ BC=a,\quad AC=b,\quad AB=c,\quad \widehat{CAB}=\alpha, \widehat{ABC}=\beta,\quad \widehat{BCA}=\gamma;\\ \widehat{ABP}=\alpha_B,\quad\widehat{ACP}=\alpha_C,\quad \widehat{BAP}=\beta_A,\quad\widehat{BCP}=\beta_C,\quad \widehat{CAP}=\gamma_A,\quad\widehat{CBP}=\gamma_B. $$ Then the initial inequality can be written as: $$ \frac{a(\sin\alpha_B+\sin\alpha_C) +b(\sin\beta_A+\sin\beta_C)+c(\sin\gamma_A+\sin\gamma_B)}{a+b+c}\le1.\tag{*} $$ which can be proved as follows: $$\begin{align} &\frac{a(\sin\alpha_B+\sin\alpha_C) +b(\sin\beta_A+\sin\beta_C)+c(\sin\gamma_A+\sin\gamma_B)}{a+b+c}\\ &=\frac{\sin\alpha(\sin\alpha_B+\sin\alpha_C) +\sin\beta(\sin\beta_A+\sin\beta_C)+\sin\gamma(\sin\gamma_A+\sin\gamma_B)} {\sin\alpha+\sin\beta+\sin\gamma}\tag1\\ &=\frac{(\sin\beta\sin\beta_A+\sin\gamma\sin\gamma_A) +(\sin\alpha\sin\alpha_B+\sin\gamma\sin\gamma_B) +(\sin\alpha\sin\alpha_C+\sin\beta\sin\beta_C)} {\sin\alpha+\sin\beta+\sin\gamma}\tag2\\ &\le\frac{\sin\alpha+\sin\beta+\sin\gamma}{\sin\alpha+\sin\beta+\sin\gamma}=1\tag3. \end{align} $$

$(1)$: Substitution: $$a=2R\sin\alpha, \quad b=2R\sin\beta, \quad c=2R\sin\gamma.$$

$(2)$: Rearrangement.

$(3)$: Transformation of sine products to cosine differences and then from cosine sums to cosine products.
$$\begin{align} &\sin\beta\sin\beta_A+\sin\gamma\sin\gamma_A\\ &=\frac{\cos(\beta-\beta_A)-\cos(\beta+\beta_A)}2 +\frac{\cos(\gamma-\gamma_A)-\cos(\gamma+\gamma_A)}2\\ &=\frac{\cos(\beta-\beta_A)+\cos(\gamma-\gamma_A)}2 -\frac{\cos(\beta+\beta_A)+\cos(\gamma+\gamma_A)}2\\ &= \underbrace{\cos\frac{\beta+\gamma-\beta_A-\gamma_A}2}_{\ge0} \underbrace{\cos\frac{\beta-\gamma-\beta_A+\gamma_A}2}_{\le1} - \underbrace{\cos\frac{\beta+\gamma+\beta_A+\gamma_A}2}_{=0}\cos\frac{\beta-\gamma+\beta_A-\gamma_A}2\tag4\\ &\le \cos\frac{\beta+\gamma-\alpha}2=\cos\left(\frac\pi2-\alpha\right)=\sin\alpha, \end{align} $$ where we used $\beta_A+\gamma_A=\alpha$.

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There may arise an interesting question when does the inequality $(\text{*})$ degenerate to equality?

It is easily seen from $(4)$ that this happens if and only if the following equations hold: $$ \begin {cases} \beta-\beta_A=\gamma-\gamma_A\\ \gamma-\gamma_B=\alpha-\alpha_B\\ \alpha-\alpha_C=\beta-\beta_C \end {cases}, $$ which together with $$ \begin {cases} \beta_A+\gamma_A=\alpha\\ \gamma_B+\alpha_B=\beta\\ \alpha_C+\beta_C=\gamma \end {cases} $$ implies: $$ \begin {cases} \beta_C=\gamma_B=\frac\pi2-\alpha\\ \gamma_A=\alpha_C=\frac\pi2-\beta\\ \alpha_B=\beta_A=\frac\pi2-\gamma \end {cases}. $$

Thus the equality holds if and only if the triangle $ABC$ is acute and $P$ is its circumcenter.

Note:

With the convention that the angles oriented oppositely to the direction towards the adjacent side are treated as negative, the inequality $(\text{*})$ becomes valid for the whole plane. Correspondingly the equality holds at the circumcenter of a triangle, regardless of its shape.