And Pythagorean Theorem, we get that the length is $\sqrt{3^2+3^2}+\sqrt{3^2+4^2}+1+\sqrt{2^2+2^2}+1=7+5 \sqrt{2}$
Search found 98 matches
- Wed Jan 11, 2017 1:48 am
- Forum: Geometry
- Topic: IGO 2016 Elementary/1
- Replies: 1
- Views: 2619
- Tue Jan 10, 2017 4:33 pm
- Forum: Geometry
- Topic: IGO 2016 Advanced/5
- Replies: 0
- Views: 2279
IGO 2016 Advanced/5
5. Do there exist six points $X_1,X_2, Y_1,Y_2,Z_1,Z_2$ in the plane such that all of the triangles $X_iY_jZ_k$ are similar for $1 \le i, j, k \le 2$?
- Tue Jan 10, 2017 4:29 pm
- Forum: Geometry
- Topic: IGO 2016 Advanced/4
- Replies: 0
- Views: 2345
IGO 2016 Advanced/4
4. In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $w_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $w_2$ is a circ...
- Tue Jan 10, 2017 4:24 pm
- Forum: Geometry
- Topic: IGO 2016 Advanced/3
- Replies: 1
- Views: 2751
IGO 2016 Advanced/3
3. Let $P$ be the intersection point of sides $AD$ and $BC$ of a convex qualrilateral $ABCD$. Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$, respectively. Let $O$ be the circumcenter of $PAB$, and $H$ the orthocenter of $PDC$. Show that the circumcircles of triangles $A...
- Tue Jan 10, 2017 4:20 pm
- Forum: Geometry
- Topic: IGO 2016 Advanced/2
- Replies: 1
- Views: 2489
IGO 2016 Advanced/2
2. In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\angle AXB = \angle DXM = 90^o$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $ \angle XMB = 2\angle MBC$.
- Tue Jan 10, 2017 4:16 pm
- Forum: Geometry
- Topic: IGO 2016 Advanced/1
- Replies: 2
- Views: 2979
IGO 2016 Advanced/1
1. Let the circles $w$ and $w'$ intersect in points $A$ and $B$. Tangent to circle $w$ at $A$ intersects $w'$ in $C$ and tangent to circle $w'$ at $A$ intersects $w$ in $D$. Suppose that the segment $CD$ intersects $w$ and $w'$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). T...
- Tue Jan 10, 2017 4:09 pm
- Forum: Geometry
- Topic: IGO 2016 Medium/5
- Replies: 1
- Views: 2597
IGO 2016 Medium/5
5. Let the circles $w$ and $w'$ intersect in points $A$ and $B$. Tangent to circle $w$ at $A$ intersects $w'$ in $C$ and tangent to circle $w'$ at $A$ intersects $w$ in $D$. Suppose that the internal bisector of $\angle CAD$ intersects $w$ and $w'$ at $E$ and $F$, respectively, and the external bise...
- Tue Jan 10, 2017 4:03 pm
- Forum: Geometry
- Topic: IGO 2016 Medium/4
- Replies: 1
- Views: 10502
IGO 2016 Medium/4
4. Let $w$ be the circumcircle of right-angled triangle $ABC (\angle A = 90)$. Tangent to $w$ at point $A$ intersects the line $BC$ in point $P$. Suppose that $M$ is the midpoint of (the smaller) arc $AB$, and $PM$ intersects $w$ for the second time in $Q$. Tangent to $w$ at point $Q$ intersects $AC...
- Tue Jan 10, 2017 3:59 pm
- Forum: Geometry
- Topic: IGO 2016 Medium/3
- Replies: 1
- Views: 2720
IGO 2016 Medium/3
3. Find all positive integers $N$ such that there exists a triangle which can be dissected into $N$ similar quadrilaterals.
- Tue Jan 10, 2017 3:57 pm
- Forum: Geometry
- Topic: IGO 2016 Medium/2
- Replies: 1
- Views: 2563
IGO 2016 Medium/2
2. Let two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. The tangent to $C_1$ at $A$ intersects $C_2$ in $P$ and the line $PB$ intersects $C_1$ for the second time in $Q$ (suppose that $Q$ is outside $C_2$). The tangent to $C_2$ from $Q$ intersects $C_1$ and $C_2$ in $C$ and $D$, respecti...