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Plane geometry problem with inequalities
ReticulatedPython 1
N
by soryn
Let 
and 
be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](//latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
and 
be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
1 reply
A well-known geo configuration revisited
Tintarn 6
N
by Primeniyazidayi
Source: Dutch TST 2024, 2.1
Let 
be a triangle with orthocenter 
and circumcircle 
. Let 
be the reflection of in and let 
the reflection of in 
. Let 
be the midpoint of segment 
. Show that the tangent to in is perpendicular to 
.
be a triangle with orthocenter 
and circumcircle 
. Let 
be the reflection of in and let 
the reflection of in 
. Let 
be the midpoint of segment 
. Show that the tangent to in is perpendicular to 
.
6 replies
AD and BC meet MN at P and Q
WakeUp 6
N
by Nari_Tom
Source: Baltic Way 2005
Let 
be a convex quadrilateral such that 
. Let and 
be the midpoints of 
and 
, respectively. The lines 
and 
meet the line 
at and 
, respectively. Prove that 
.
be a convex quadrilateral such that 
. Let and 
be the midpoints of 
and 
, respectively. The lines 
and 
meet the line 
at and 
, respectively. Prove that 
.
6 replies
Angle EBA is equal to Angle DCB
WakeUp 6
N
by Nari_Tom
Source: Baltic Way 2011
Let be a convex quadrilateral such that 
. Suppose that a point on the side satisfies the equality
![\[AE\cdot ED + BE^2=CD\cdot AE.\]](//latex.artofproblemsolving.com/d/7/5/d75cc3a449a56b43eac574cc44e9932a2e63b33b.png)
Show that
.
be a convex quadrilateral such that 
. Suppose that a point on the side satisfies the equality![\[AE\cdot ED + BE^2=CD\cdot AE.\]](http://latex.artofproblemsolving.com/d/7/5/d75cc3a449a56b43eac574cc44e9932a2e63b33b.png)
Show that
.
6 replies
Reactangle
Mathx 3
N
by Nari_Tom
Source: Baltic Way 2003
In a rectangle be a rectangle and 
, let be the midpoint of and an arbitrary inner point of . Let 
and 
be the feet of perpendiculars drawn correspondingly from to 
and from to 
. Prove that the points 
are concyclic.
be a rectangle and 
, let be the midpoint of and an arbitrary inner point of . Let 
and 
be the feet of perpendiculars drawn correspondingly from to 
and from to 
. Prove that the points 
are concyclic.
3 replies
Unit square cut into m quadrilaterals inequality
WakeUp 2
N
by Nari_Tom
Source: Baltic Way 2009
A unit square is cut into 
quadrilaterals 
. For each 
let 
be the sum of the squares of the four sides of 
. Prove that
![\[S_1+\ldots +S_m\ge 4\]](//latex.artofproblemsolving.com/c/9/d/c9d3ae0dd89043c34e5c4214c22398ad6fec3a7d.png)
quadrilaterals 
. For each 
let 
be the sum of the squares of the four sides of 
. Prove that![\[S_1+\ldots +S_m\ge 4\]](http://latex.artofproblemsolving.com/c/9/d/c9d3ae0dd89043c34e5c4214c22398ad6fec3a7d.png)
2 replies
Projection of M onto the line l is on PK
WakeUp 3
N
by Nari_Tom
Source: Baltic Way 2009
Let be the midpoint of the side 
of a triangle , and let 
be a point on the ray 
beyond . The line 
intersects the side at the point 
. is the point on the segment 
such that 
is the bisector of the angle 
. The line 
passes through and is parallel to . Prove that the projection of the point onto the line belongs to the line 
.
be the midpoint of the side 
of a triangle , and let 
be a point on the ray 
beyond . The line 
intersects the side at the point 
. is the point on the segment 
such that 
is the bisector of the angle 
. The line 
passes through and is parallel to . Prove that the projection of the point onto the line belongs to the line 
.
3 replies
Two circles
k2c901_1 11
N
by Primeniyazidayi
Source: Taiwan NMO 2006
are two fixed points on a circle centered at 
, and is an interior point of the circle that differs from . 
are concyclic. Prove that the bisector of 
is perpendicular to line 
.
11 replies
Concyclic points
socrates 6
N
by Nari_Tom
Source: Baltic Way 2014, Problem 14
Let be a convex quadrilateral such that the line 
bisects the angle 
The circumcircle of triangle intersects the sides and in the points and 
respectively. The line through and parallel to intersects the lines and at the points 
and 
respectively. Prove that the points 
and 
lie on a common circle.
be a convex quadrilateral such that the line 
bisects the angle 
The circumcircle of triangle intersects the sides and in the points and 
respectively. The line through and parallel to intersects the lines and at the points 
and 
respectively. Prove that the points 
and 
lie on a common circle.
6 replies
Triangles with equal areas
socrates 11
N
by Nari_Tom
Source: Baltic Way 2014, Problem 13
Let be a square inscribed in a circle 
and let be a point on the shorter arc of . Let 
and 
Show that triangles
and 
have equal areas.
be a square inscribed in a circle 
and let be a point on the shorter arc of . Let 
and 
Show that triangles
and 
have equal areas.
11 replies
S must be the incentre of triangle ABC
WakeUp 3
N
by Nari_Tom
Source: Baltic Way 2007
In triangle let 
and 
be the altitudes. Let the points 
and fulfil the following requirements:
i) is the circumcentre of triangle .
ii) All the segments
and 
are equal to the circumradius of triangle .
iii) The oriented segment
has the same direction as the oriented segment . Similarly, 
has the same direction as 
, and 
has the same direction as .
Prove that is the incentre of triangle .
let 
and 
be the altitudes. Let the points 
and fulfil the following requirements:i)
is the circumcentre of triangle .ii) All the segments
and 
are equal to the circumradius of triangle .iii) The oriented segment
has the same direction as the oriented segment . Similarly, 
has the same direction as 
, and 
has the same direction as .Prove that
is the incentre of triangle .
3 replies
