


Points, Theorems and Problems  Index.


Perpendicular
Bisector. 

Butterfly Theorem
Proof with animation.
Midpoint of a chord. 

Median of a Trapezoid, Theorems and Problems.
Index.


Newton's
Theorem: Newton's Line. Circumscribed quadrilateral, midpoints of
diagonals, center of the circle inscribed.


GeoGebra, Dynamic Geometry:
Nine Point Circle
HTML5 Animation for Tablets (iPad, Nexus..) 

GeoGebra, Dynamic Geometry:
Internal Feuerbach Point
HTML5 Animation for Tablets (iPad, Nexus..) 

Equilic Quadrilateral.
120 degrees. Midpoints. 

Kurschak's Tile and Theorem.
Jozsef Kurschak (Hungary, 18641933) An elegant and a purely geometric
way of finding the area of a regular dodecagon.
Midpoints. 

Varignon and Wittenbauer parallelograms. Quadrilateral: midpoints and
trisection points of the edges.


Median
length, Apollonius' Theorem
Midpoints. 

Van Aubel's Theorem.
Quadrilateral with Squares. Proof with animation.
Midpoints. 

Symmedian (Lemoine)
Point
Triangle Center 

Archimedes' Book of Lemmas:
Proposition 7
Square and inscribed and circumscribed Circles Midpoints.


Archimedes' Book of Lemmas:
Proposition 2
Semicircle, diameter, tangents, secant line.
Midpoint. 

Archimedes' Book of Lemmas:
Proposition 3
Semicircle, perpendicular to diameter, chords.
Midpoint. 

Archimedes' Book of Lemmas:
Proposition 10
Tangents to a circle, secant line and chords, Parallel.
Midpoint. 

Archimedes' Book of Lemmas:
Proposition 15
Circle, Diameter, Inscribed Regular Pentagon, Arc, Midpoint,
Perpendicular, Radius. 

Bottema's Theorem:
Triangle and Squares with Interactive Geometry Software
StepbyStep construction, Manipulation, and animation.
Midpoints. 

EightPoint Circle Theorem
StepbyStep construction, Manipulation, and animation.
Perpendiculars from the midpoints. 

Marion
Walter's Theorem: Triangle and Hexagon areas:
Using TracenPoche Dynamic Geometry Software.
Midpoints. 

NinePoint Center, NinePoint Circle, Euler Line.
Interactive illustration.


Mascheroni
Construction with compass alone. Midpoint of a segment.


Heron's Formula with medians.
Midpoints. 

Mittenpunkt Point
Midpoints. 

Dynamic Geometry: Brahmagupta Theorem,
. GeoGebra, HTML5 Animation for Tablets 

Geometry Problem 1575:
Prove an Angle Bisector in a Triangle Involving an Altitude, Midpoint, and Excircle 

Geometry Problem 1564:
Find the Area of Quadrilateral BGDJ in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. 

Geometry Problem 1563:
Perpendicularity in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints. 

Geometry Problem 1562:
Proof of Collinearity in a Rightangled Triangle involving the Altitude, Angle bisectors, and Midpoint. 

Geometry Problem 1560:
Trapezoid ABCD: Unlocking Angle Secrets at G and H. 

Geometry Problem 1559:
Proving BC Bisects Segment DE. This geometry problem challenges students at high school and college levels. 

Geometry Problem 1558:
The midpoints of segments connecting corresponding vertices of equilateral triangles form an equilateral triangle. 

Geometry Problem 1557:
Calculate the Angle DHG in Right Triangle ABC. The problem involves: equilateral,
isosceles, midpoint and congruence 

Geometry Problem 1554: Finding the Length of Side AB in Triangle ABC 

Geometry Problem 1553: Solving for OC in Triangle ABC with Unique Angle Bisectors 

Geometry Problem 1548: Exploring Segment DE Length in Triangle ABC with a 45Degree Angle and Intersecting Altitudes 

Geometry Problem 1543: Calculating the Area of Quadrilateral ABED in a Square with a Side Length of 20 and an Intersecting Arc. 

Geometry Problem 1539 Demystified: Unraveling the Lengths in an Isosceles Triangle with Altitude and Tangent Secrets!. 

Geometry Problem 1538: Unlocking the Secrets of Triangular Geometry: Solve for the Area of a Quadrilateral Using External Squares and a Segment Length. 

Geometry Problem 1537 Challenge: Can You Solve for the Missing Area in a Parallelogram using Midpoints and Intersection Points?. 

Geometry
Problem 1536: Discover the Power of Midpoints: Solving for Missing Areas in Quadrilaterals. 

Geometry
Problem 1525 and a Thematic Poem.
Unveiling the Secrets of an Equilateral Triangle in Right Triangle Geometry: Finding the Midpoint Distance between Segments. 

Geometry
Problem 1524 and a Thematic Poem.
Unlock the Mystery of Parallelograms: Discover the Length of Segment between the Intersecting Angle Bisectors. 

Geometry
Problem 1523 and a Thematic Poem.
Discover How to Calculate the Length of the Altitude in an Isosceles Triangle  Get Expert Geometry Tips Now!
. 

Geometry
Problem 1522 and a Thematic Poem.
Unlocking the Angle Measure of a Triangle with Median and Doubled Side Lengths. 

Geometry
Problem 1519 and a Thematic Poem.
Discover the Length of a Segment in a Parallelogram using Midpoints and Parallel Lines. 

Geometry
Problem 1517 and a Thematic Poem.
Unlocking Triangle Side Length: Solving with a Median and Two Angles. Difficulty Level: High School. 

Geometry
Problem 1515 and a Thematic Poem.
Mastering Triangle Distance Calculation: Find the Distance from the Intersection of Medians to an Exterior Line. Difficulty Level: High School. 

Geometry
Problem 1512 and a Thematic Poem.
Finding the Length of a Segment in a Triangle with a Median and a Cevian with Given Ratio. Difficulty Level: High School. 

Geometry
Problem 1494.
Parallelogram, Midpoints, Octagon, Areas. 

Dynamic Geometry 1476.
DrozFarny Line Theorem, Triangle, Orthocenter, Perpendicular, Collinear Midpoints, Stepbystep Illustration. GeoGebra,
iPad. 

Dynamic Geometry 1474.
Butterfly Theorem, Circle, Chords, Midpoints, Stepbystep Illustration. 

Dynamic Geometry 1469.
Triangle, Circumradius, Inradius, Midpoints, Arcs, Sum of Distances, Stepbystep Illustration. 

Geometry Problem 1466.
Tangential Quadrilateral, Newton Line, Incenter, Midpoint, Diagonal, Collinear Points.
Stepbystep Illustration using GeoGebra. 

Geometry Problem 1464.
Quadrilateral, Interior Point, Midpoint of Sides, Equal Sum of Areas.
Stepbystep Illustration using GeoGebra. 

Geometry Problem 1462.
NewtonLine, NewtonGauss Line, Complete Quadrilateral, Midpoints of Sides and Diagonals,
Collinear Points. 

Dynamic Geometry
1460.
NewtonGauss Line, Complete Quadrilateral, Midpoints of Diagonals,
Collinear Points, Stepbystep Illustration using GeoGebra. 

Geometry
Problem 1459.
Two Triangles, Orthocenter, Midpoint, Perpendicular, Stepbystep Illustration using GeoGebra. 

Dynamic Geometry Problem 1444.
The Asymmetric Propeller Theorem, Equilateral Triangles, Midpoints. Using GeoGebra. 

Geometry Problem 1429. Triangle, Median, 1530 degree, Angle, Congruence. 

Geometry Problem 1417. Triangle, Cevian, Circumcircle, Incenter, Perpendicular, Angle Bisector,
Tangent Lines, Tangent Circles, Midpoint. 

Geometry Problem 1408. Right Triangle, Incircle, Excircle, Incenter, Midpoint, Tangency Point, Collinearity. 

Geometry Problem 1395. Triangle with three rectangles on the sides, Midpoints, Perpendicular lines, Concurrency. 

Geometry Problem 1385. Triangle, Orthocenter, Circle, Circumcircle, Angle Bisector, Midpoint, Collinear Points. 

Geometry Problem 1365.
Circle, Tangent, Secant, Chord, Parallel, Midpoint, Sketch, iPad Apps. 

Geometry Problem 1360.
Triangle, NinePoint Circle, Feuerbach's Circle, Euler's Circle, Cyclic Quadrilateral, Concyclic Points, Sketch, iPad Apps. 

Geometry Problem 1359.
Triangle, Circumcircle, Congruent Angles, Midpoints. 

Geometry Problem 1355.
Triangle, Midpoint, Median, Circle, Chord, Equal Product of measure of segments. 

Geometry Problem 1351. Two Tangential or Circumscribed Quadrilaterals, Midpoints, Perpendiculars. 

Problem 1342.
Circle, Secant, Chord, Midpoint, Concyclic Points, Cyclic Quadrilateral.


Problem 1336.
Right Triangle, Cevian, Double Angle, Perpendicular, 90 Degrees.


Geometry Problem 1325.
Octagons, Midpoints, Squares, Mobile Apps, iPad. 

Geometry Problem 1317.
Triangle, Excircle, Chord, Tangent, Midpoint, Arc, Sum of two Segments, Congruence. 

Geometry Problem 1316.
Triangle, Incircle, Tangent, Chord, Midpoint, Sum of two Segments, Congruence. 

Geometry Problem 1302.
Triangle, Median, Midpoint, Circles, Diameters, Congruence. 

Geometry Problem 1301.
Arbelos, Semicircles, Diameters, Circle, Incircle, Incenter, Square, Midpoint, Concurrency. 

Geometry Problem 1298.
Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Angle Bisector, Perpendicular, Midpoint. 

Geometry Problem 1294.
Triangle, Altitudes, Four Squares, Center, Midpoint, Concyclic Points. 

Geometry Problem 1293.
Triangle, Altitudes, Two Squares, Center, Concurrent Lines, Midpoint. 

Geometry Problem 1291.
Three Squares, Collinear Vertices, Midpoint, Congruence. 
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