Found problems: 14
Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.
It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.
Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.
[i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$.
[i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then
\[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\]
where $r_1$ is the exradius corresponding to the vertex $A.$
In triangle $ABC$ we have $AC = CB$. On side $AB$ is a point $D$ such that the radius of the incircle of triangle $ACD$ is equal to the radius of the circle tangent to the segment $DB$ and to the extensions of the lines $CD$ and $CB$. Prove that this radius equals a quarter of either of the two equal altitudes of triangle $ABC$.
A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.
[i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$.
[i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then
\[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\]
where $r_1$ is the exradius corresponding to the vertex $A.$
In all tetrahedron $ABCD$ holds
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[*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$
[*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$
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for all $t\in [0,1]$
In all tetrahedron $ABCD$ holds
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[*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$
[*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$
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for all $n\in \mathbb{N}^*$
In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that
$$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$
Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one
A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that:
a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$.
b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.
The radius of the circle inscribed in triangle $ABC$ is equal to $r$, and the radius of the circle tangent to the segment $BC$ and the extensions of sides $AB$ and $AC$ (the exscribed circle corresponding to angle $A$) is equal to $R$. A circle with radius $x < r$ is inscribed in angle $\angle BAC$. Tangents to this circles passing through points $B$ and $C$ and different from $BA$ and $AC$ intersect at point $A'$. Let $y$ be the radius of the circle inscribed in triangle $BCK$. Find the greatest value of the sum $x + y$ as x changes from $0$ to $r$. (In this case, it is necessary to prove that this largest value is the same in any triangle with given $r$ and $R$).
Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$