Search found 155 matches
- Sun Nov 22, 2015 1:18 am
- Forum: Secondary Level
- Topic: geometry
- Replies: 3
- Views: 13738
Re: geometry
$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OB||AC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$
- Wed May 20, 2015 12:51 am
- Forum: Geometry
- Topic: Geometric Inequality
- Replies: 1
- Views: 2934
Re: Geometric Inequality
Any hints? I'm stuck...
- Mon May 04, 2015 8:39 pm
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies: 5
- Views: 5375
Re: Inscribed-Quad in an Excribed-Quad
Extremely sorry for the typo...
- Sun May 03, 2015 2:35 pm
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies: 5
- Views: 5375
Inscribed-Quad in an Excribed-Quad
The quadrilateral $ABCD$ is excribed around a circle with centre $I$. Prove that, the projections of $B$ and $D$ on $IA$, $IC$ lie on a circle.
Sharygin Geometry Olympiad, Russia
Sharygin Geometry Olympiad, Russia
- Sun May 03, 2015 1:43 am
- Forum: Geometry
- Topic: triangular inequality [sides and area]
- Replies: 3
- Views: 3889
Re: triangular inequality [sides and area]
This was also set as Problem No. 2 in the 1961 IMO.
- Wed Apr 29, 2015 9:39 pm
- Forum: Algebra
- Topic: Inequality (sin, r and s)
- Replies: 2
- Views: 3365
Inequality (sin, r and s)
Prove the inequality,
\[\frac{1}{\sqrt{2\sin A}}+\frac{1}{\sqrt{2\sin B}}+\frac{1}{\sqrt{2\sin C}}\leq \sqrt{\frac{s}{r}}\]
where, $s$ and $r$ are the semi-perimeter and inradius of $\triangle ABC$.
Sharygin Geometry Olympiad, Russia
\[\frac{1}{\sqrt{2\sin A}}+\frac{1}{\sqrt{2\sin B}}+\frac{1}{\sqrt{2\sin C}}\leq \sqrt{\frac{s}{r}}\]
where, $s$ and $r$ are the semi-perimeter and inradius of $\triangle ABC$.
Sharygin Geometry Olympiad, Russia
Re: cool geo
Corrected.
Re: cool geo
$AEHF$ is cyclic. $\therefore P \in \odot AEF$ iff $\angle APH= \angle AFH =90^{\circ}$. Suppose, $AP\cap\odot ABC=G$. Since, $G$ is the mid-point of arc $\widehat{BC}$, $M$ is the mid-point of $OG$. Now, $AN||OM$ and $AN=OM$ $\Rightarrow AOMN$ is a prallelogram $\Rightarrow AO||MN$ $\Rightarrow \tr...
- Tue Feb 24, 2015 10:15 pm
- Forum: Higher Secondary Level
- Topic: Vectors around Regular Polygon
- Replies: 4
- Views: 13222
Re: Vectors around Regular Polygon
Hmm, Nayel Vai... Complex Numbers definitely give a straightforward solution. But, Mahi's approach was very neat. :) Solution using complex numbers: Let, the polygon be inscribed in the unit-circle, and, $OP_1$ be the real axis. Then, \[\overrightarrow{OP_i}=\cos\frac{2\pi}{n}(i-1)+i\cdot\sin\frac{2...
- Sat Feb 21, 2015 11:25 pm
- Forum: Higher Secondary Level
- Topic: Vectors around Regular Polygon
- Replies: 4
- Views: 13222
Vectors around Regular Polygon
Let, $ P_1P_2P_3\ldots P_n$ be a regular polygon whose circumcentre is $O$. Prove that,
\[\sum_{i=1}^n \overrightarrow{OP_i}=0\]
\[\sum_{i=1}^n \overrightarrow{OP_i}=0\]