Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1960 IMO Problems/Problem 4"

(New page: ==Problem== ==Solution== {{solution}} ==See Also== {{IMO box|year=1960|num-b=3|num-a=5}})
 
m
 
(7 intermediate revisions by 4 users not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 +
Construct triangle <math>ABC</math>, given <math>h_a</math>, <math>h_b</math> (the altitudes from <math>A</math> and <math>B</math>), and <math>m_a</math>, the median from vertex <math>A</math>.
  
 
==Solution==
 
==Solution==
  
−
{{solution}}
+
Let <math>M_a</math>, <math>M_b</math>, and <math>M_c</math> be the midpoints of sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively. Let <math>H_a</math>, <math>H_b</math>, and <math>H_c</math> be the feet of the altitudes from <math>A</math>, <math>B</math>, and <math>C</math> to their opposite sides, respectively. Since <math>\triangle ABC\sim\triangle M_bM_aC</math>, with <math>M_bM_a=\frac12 AB</math>, the distance from <math>M_a</math> to side <math>\overline{AC}</math> is <math>\frac{h_b}{2}</math>.
 +
 
 +
Construct <math>AM_a</math> with length <math>m_a</math>. Draw a circle centered at <math>A</math> with radius <math>h_a</math>. Construct the tangent <math>l_1</math> to this circle through <math>M_a</math>. <math>\overline{BC}</math> lies on <math>l_1</math>.
 +
 
 +
Draw a circle centered at <math>M_a</math> with radius <math>\frac{h_b}{2}</math>. Construct the tangent <math>l_2</math> to this circle through <math>A</math>. <math>\overline{AC}</math> lies on <math>l_2</math>. Then <math>C=l_1\cap l_2</math>.
 +
 
 +
Construct the line <math>l_3</math> parallel to <math>l_2</math> so that the distance between <math>l_2</math> and <math>l_3</math> is <math>h_b</math> and <math>M_a</math> lies between these lines. <math>B</math> lies on <math>l_3</math>. Then <math>B=l_1\cap l_3</math>.
 +
 
 +
==Video Solution==
 +
 
 +
https://youtu.be/M0_UdvxH890
  
 
==See Also==
 
==See Also==
  
−
{{IMO box|year=1960|num-b=3|num-a=5}}
+
{{IMO7 box|year=1960|num-b=3|num-a=5}}
 +
[[Category:Olympiad Geometry Problems]]
 +
[[Category:Geometric Construction Problems]]

Latest revision as of 00:43, 2 January 2023

Problem

Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

Solution

Let $M_a$, $M_b$, and $M_c$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Let $H_a$, $H_b$, and $H_c$ be the feet of the altitudes from $A$, $B$, and $C$ to their opposite sides, respectively. Since $\triangle ABC\sim\triangle M_bM_aC$, with $M_bM_a=\frac12 AB$, the distance from $M_a$ to side $\overline{AC}$ is $\frac{h_b}{2}$.

Construct $AM_a$ with length $m_a$. Draw a circle centered at $A$ with radius $h_a$. Construct the tangent $l_1$ to this circle through $M_a$. $\overline{BC}$ lies on $l_1$.

Draw a circle centered at $M_a$ with radius $\frac{h_b}{2}$. Construct the tangent $l_2$ to this circle through $A$. $\overline{AC}$ lies on $l_2$. Then $C=l_1\cap l_2$.

Construct the line $l_3$ parallel to $l_2$ so that the distance between $l_2$ and $l_3$ is $h_b$ and $M_a$ lies between these lines. $B$ lies on $l_3$. Then $B=l_1\cap l_3$.

Video Solution

https://youtu.be/M0_UdvxH890

See Also

1960 IMO (Problems)
Preceded by
Problem 3 		1 • 2 • 3 • 4 • 5 • 6 • 7 Followed by
Problem 5
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1960_IMO_Problems/Problem_4&oldid=185476"