Machu Picchu, Geometry Theorems and Problems, Online Education

Geometry Theorems and Problems - Recent Additions (Page 9 of 15)

Euclidean Geometry

Triangle, Cevian, Angle, 90 degree, Congruence

Proposed Problem 424.
Triangle, Cevian, Angle, 90 degree, Congruence.

a

Euclid's Elements Book I, Proposition 4: (Side-Angle-Side SAS)

Euclid's Elements Book I, Proposition 3

Euclid's Elements Book I, Proposition 3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less

Euclid's Elements Book I, Proposition 2

Euclid's Elements Book I, Proposition 2: To place at a given point (as an extremity) a straight line equal to a given straight line

Euclid's Elements Book I, Proposition 1

Euclid's Elements Book I, Proposition 1: On a given finite line to construct an equilateral triangle

Triangle, Angle, Congruence

Proposed Problem 423.
Triangle, 30, 40 degree, Angle, Congruence.

Triangle, Angle, Congruence

Proposed Problem 422.
Triangle, 24, 30 degree, Angle, Congruence.

Right triangle, Angles, Measurement

Proposed Problem 421.
Right Triangle, Cevian, Angles, Measurement.

Triangle, Angle, Altitude, Side, Measurement

Proposed Problem 420.
Triangle, Altitude, Angles, Sides, Measurement.

Triangle, Incircle, Excircle, Inradius, Exradius

Proposed Problem 419.
Triangle, Cevian, Incircle, Excircle, Inradius, Exradius

Triangle, Incircle, Inradius, Tangent

Proposed Problem 418.
Triangle, Incircle, Inradius, Equal Tangent circles, Radius.

Tangent circles, Angles

Proposed Problem 417.
Tangent circles, Tangent lines, Angles.

Routh's theorem index

Routh's Theorem - Index
Triangle, Cevians, Area, Ratio.
 

Routh's Theorem 1

Routh's Theorem 1.
Triangle, Cevians, Ratio of Areas.

Routh's theorem: Triangle, Cevians, Area Ratio

Routh's Theorem 2.
Triangle, Cevians, Ratio of Areas.

ROuth's Theorem 3: Triangle, Cevians, Ratio Area

Routh's Theorem 3.
Triangle, Cevians, Ratio of Areas.

Routh's theorem 4: Triangle, Cevians, Area

Routh's Theorem 4.
Triangle, Cevians, Ratio of Areas.

Area of a triangle, Cevian, Trisector

Proposed Problem 416.
Triangle, Cevians, Trisectors, Area.

Right triangle, Cevian, Angle, Congruence

Proposed Problem 415.
Right Triangle, Cevian, Angles, Congruence.

Triangle, Angle, Median, Altitude, Congruence

Proposed Problem 414.
Triangle, Angles, Altitude, Median, Congruence.

Cyclic quadrilateral, Orthocenter, Parallelogram, Congruence

Proposed Problem 413.
Cyclic Quadrilateral, Orthocenter, Parallelogram, Concurrency, Congruence.

Triangle, Angles, Congruence

Proposed Problem 412.
Triangle, Angles, Congruence.

Triangle, Median, Angles

Proposed Problem 411.
Triangle, Median, Angles.

Regular pentagons, Angle

Proposed Problem 410.
Two Regular Pentagons, Angle.

Quadrilateral, Angles, Diagonals, Perpendicular

Proposed Problem 409.
Quadrilateral, Diagonals, Perpendicular, Congruence.

Cyclic quadrilateral, Perpendicular, Parallelogram, Congruence

Proposed Problem 408.
Cyclic quadrilateral, Perpendicular, Parallelogram, Congruence.

Right triangle, 15 degrees

Proposed Problem 406.
Right triangle, 15 degrees, Midpoints, Congruence.

Quadrilateral, 60, 75, 135 degrees

Proposed Problem 405.
Quadrilateral, 60, 75, and 135 degrees, Midpoint.

Concurrent lines, Index

Concurrent lines.
Theorems and Problems Index.

Equilateral triangles

Proposed Problem 404.
External Equilateral triangles, Congruent and Concurrent Lines.

Trapezoid median, Theorems and Problems, Index

Median of a Trapezoid, Theorems and Problems. Index.

Circular Sector Index

Circular Sector. Index.
Theorems and Problems. 

Circular Sector 90 degrees

Proposed Problem 403.
Circular sector, 90 degrees, Circle, Semicircle, Area.

Hexagon, Centroids

Proposed Problem 402.
Hexagon, Triangle, Centroid, Parallel, Congruence, Similarity.

Area of a triangle, Circumcircle, Angle Bisector

Proposed Problem 401.
Area, Triangle, Angle bisector, Circumcircle, Perpendicular bisector, Congruence.

Point, Theorems and Problems, Index

Points, Theorems and Problems - Index.

Triangle, Angle Bisector, Circumcircle

Proposed Problem 400.
Triangle, Angle bisector, Circumcircle, Perpendicular, Congruence.

RIght triangle, Midpoints, Distance

Proposed Problem 399.
Right triangle, Midpoints, Distance, Pythagorean theorem, Congruence.

Parallelogram, Area

Proposed Problem 398.
Parallelogram, Area, Similarity, Congruence.

Triangle, Altitude

Proposed Problem 397.
Triangle, Altitude, Midpoints, Congruence.

 

Square, Angle trisectors

Proposed Problem 396.
Square, Angle Trisectors, Congruence, Area.

Square, 15 degree

Proposed Problem 395.
Square, 15 Degree, Equilateral triangle.

Kosnita theorem

Kosnita's Theorem.
Circumcenters, Concurrent lines.

Square, Circle

Proposed Problem 394.
Square, 90 Degree Arc, Diagonal, Congruence.

Triangle, Orthocenter

Proposed Problem 393.
Triangle, Orthocenter, Circumcircles, Congruence, Collinear.

Median of a triangle, Index

Median Index.
Theorems and Problems.

Triangle, Parallel, Concurrent lines

Proposed Problem 392.
Triangle, Parallel lines, Concurrent lines.

Triangle, Parallel, Collinear points

Proposed Problem 391.
Triangle, Parallel lines, Collinear points.

Triangle, Parallel, Collinear points

Proposed Problem 390.
Triangle, Parallel lines, Collinear points.

Triangle, Parallel, Harmonic Mean

Proposed Problem 389.
Triangle, Parallel lines, Harmonic Mean.

Triangle, Angle bisector, Perpendicular, Parallel

Proposed Problem 388.
Triangle, Angle bisector, Perpendicular, Parallel.

Triangle, Angle Bisector, Perpendicular, Concyclic points

Proposed Problem 387.
Triangle, Angle bisector, Perpendicular, Concyclic points.

Triangle, Angle bisectors

Proposed Problem 386.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter, Congruence.

Triangle, Angle bisectors

Proposed Problem 385.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter.

Triangle, Angle bisectors

Proposed Problem 384.
Triangle, Angle bisector, Perpendicular, Parallel, Semiperimeter.

Concave Quadrilateral, Angle bisectors

Proposed Problem 383.
Concave Quadrilateral, Angle Bisectors, Semi-difference of Angles.

Quadrilateral, Angle Bisectors

Proposed Problem 382.
Concave Quadrilateral, Angle Bisectors, Angles.

Quadrilateral, Angle Bisectors

Proposed Problem 381.
Quadrilateral, Diagonals, Angle Bisectors, Angles.

Chord Index

Chord. Index.
A chord of a circle is the line segment joining two points on the circle. 

Triangle, Excenter, Parallel

Proposed Problem 380.
Triangle, Excenter, Parallel to a side, Angle, Congruence.

Triangle, Excenter, Parallel

Proposed Problem 379.
Triangle, Excenter, Parallel to a side, Angle, Congruence.

Triangle, Incenter, Parallel to a side

Proposed Problem 378.
Triangle, Incenter, Parallel to a side, Angle, Congruence.

Triangle, Altitude, Angle bisector

Proposed Problem 377.
Triangle, Internal Angle Bisector, Altitude.

Triangle, Excenter, Internal and External angle bisector,

Proposed Problem 376.
Triangle, Excenter, Internal and External Angle Bisectors.

Triangle, Excenter, Angle

Proposed Problem 375.
Triangle, Excenter, External Angle Bisectors.

Tiangle, Incenter, Angle

Proposed Problem 374.
Triangle, Incenter, Internal Angle Bisectors.

Problem about square and inscribed circle

Proposed Problem 373.
Square, Inscribed Circle, Diagonal, Perpendicular, Angle.

Circles, Common tangent

Proposed Problem 372.
Circles, Common Internal and External Tangent, Angles.

Square, Triangle, Area, Circle

Proposed Problem 371.
Square, Inscribed circle, Triangle, Area.

Triangle with squares, Circumcircles

Proposed Problem 370.
Triangle with squares, Circumcircles, Tangent circles.

Intersecting circles, chord, angle, congruence 

Proposed Problem 369.
Intersecting circles, Chord, Center, Angle, Congruence.

Triangle, 120 degrees, angle bisectors

Proposed Problem 368.
Triangle, 120 degrees, Angle bisectors, Perpendicular.

Square, 45 degrees

Proposed Problem 367.
Square, Angle, 45 degrees, Pythagoras, Congruence, Similarity.

Equilateral triangle, circumcircle

Proposed Problem 366.
Scalene triangle, Circumcircle, Angles, 60 Degrees, Equilateral triangle.

Circular sector 60 degrees

Proposed Problem 365.
Circular Sector of 60 degrees, Midpoints, Perpendicular, Congruence.

Triangle area, Circle, Curvilinear triangle

Proposed Problem 364.
Obtuse triangle, Congruence, Circles, Diameter, Curvilinear triangle, Area.

Special Right Triangle 30-60 Index

Special Right Triangle 30-60.

Special right triangle 30 60

Special Right Triangles.

Obtuse Triangle Index

Obtuse Triangle.

Right triangle, Congruence, Angles

Proposed Problem 363.
Right triangle, Congruence, Angles.

Circle, Chord, Perpendicular

Proposed Problem 362.
Circle, Chord, Perpendicular, Equal chords.

Euclid's Elements Book I

Euclid's Elements Book I Word Cloud.

Macheroni Constructions Index

Mascheroni construction with compass alone. Index

Right triangle, Incircle, Angle

Proposed Problem 361.
Right triangle, Incircle, Incenter, Tangency points, Angle.

Area of Triangle, Circles, Diameters

Proposed Problem 360.
Area of a triangle, Curvilinear triangles, Circles, Diameters.

Cyclic quadrilateral, Rhombus

Proposed Problem 359.
Cyclic quadrilateral, Angle bisector, Rhombus.

Mascheroni Construction with compass alone

Mascheroni construction: Find the center of a circle with compass alone.

Isosceles triangle 80 80 20

Proposed Problem 358.
Isosceles triangle 80-80-20, Circle, Angles, Congruence.

Euclid Elements Book X, Lemma for Proposition 33

Euclid's Elements, Book X, Lemma for Proposition 33 One page visual illustration.

Square, Exterior point, Triangle, Area

Proposed Problem 357.
Square, Exterior point, Triangles, Area.

Square, Triangle, Area

Proposed Problem 356.
Square, Point inside, Triangles, Area.

Circles, Tangent, Concyclic points

Proposed Problem 355.
Circles, Common external tangent, Concyclic points.

Rhombus, Square 

Proposed Problem 354.
Rhombus, Square, 45 degrees.

Tangential quadrilateral, Incircle, Inradius

Proposed Problem 353.
Tangential quadrilateral, Diagonals, Incircles, Inradii.

Tangential quadrilateral

Proposed Problem 352.
Tangential quadrilateral, Incircles, Common tangent, Circumscribable or Tangential quadrilateral.

Rhombus, Circumscribable quadrilateral

Proposed Problem 351.
Rhombus, Incircles, Common tangent, Circumscribable or Tangential quadrilateral.

Triangle, Cevian, Incircles, Angles

Proposed Problem 350.
Triangle, Cevian, Incircles, Tangents, Tangency Point, Angles.

Triangle, Cevian, Incircles

Proposed Problem 349.
Triangle, Cevian, Incircles, Tangents, Tangency Point.

Common tangents

Proposed Problem 348.
Circles, Common External Tangents, Common Internal Tangent.

Triangle, Altitude, Circle, Concyclic points

Proposed Problem 347.
Triangle, Altitude, Perpendicular, Circle, Concyclic points.

Geometry Problem: Equal circles, Tangents, Hexagon

Proposed Problem 346.
Equal circles, Tangents, Hexagon, Semiperimeter.

Geometry Problem: Equal circles, Tangents

Proposed Problem 345.
Equal circles, Tangents, Concurrent lines, Hexagon, Semiperimeter.

Quadrilateral, Circle, Tangent

Proposed Problem 344.
Quadrilateral, Extension of sides, Circle, Tangent, Semiperimeter.

Hexagon, Inscribed Circle

Proposed Problem 343.
Hexagon, Inscribed circle, Tangent, Semiperimeter.

Pentagon, Circle, Tangent,Semiperimeter

Proposed Problem 342.
Pentagon, Inscribed circle, Tangent, Semiperimeter.

Triangle, inscribed circle, tangent

Proposed Problem 341.
Triangle, Inscribed circle, Tangent, Semiperimeter.

Triangle, Angle Bisectors

Proposed Problem 340.
Triangle, Angle Bisectors, Perpendiculars, Distances. Exterior Point.

Triangle, Angle Bisectors

Proposed Problem 339.
Triangle, Angle Bisectors, Perpendiculars, Distances. Interior Point.

Problem 338 concyclic points

Proposed Problem 338.
Triangle, Circumcircle, Inscribed Circle, Exterior angle bisector, Concyclic points.

Isosceles Trapezoid, Concyclic Points

Proposed Problem 337.
Isosceles Trapezoid, Angle bisector, Parallel, Concyclic points.

Tangent Circles Index

Tangent Circles Index
 

Tangent circles, Square

Proposed Problem 336.
Two equal circles, a Common Tangent and a Square.

Cyclic Quadrilateral, Perpendiculars to Sides

Proposed Problem 335.
Cyclic Quadrilateral, Perpendiculars to Sides.

Problem 334 Cyclic Quadrilateral, Perpendiculars to Diagonals

Proposed Problem 334.
Cyclic Quadrilateral, Perpendiculars to Diagonals.

Midpoint Theorems and Problems Index

Midpoints.
Theorems and Problems

Circle and Inscribed Semicircle

Proposed Problem 333.
Circle inscribed in a semicircle, Perpendicular to the common tangent.

Cyclic Quadrilateral Problem

Proposed Problem 332.
Cyclic quadrilateral, Ratio of diagonals, Similarity.

Elearning 331 Square, Inscribed Circle

Proposed Problem 331.
Square, Point on the Inscribed Circle, Tangency Points.

Cyclic quadrilateral

Proposed Problem 330.
Cyclic quadrilateral, Perpendicular diagonals, Area, Circumcenter.

Triangle, Altitudes, Concyclic Points

Proposed Problem 329.
Triangle, Altitudes, Circle, Diameter, Concyclic points.

Triangle, Incircle, Parallel, Midpoint

Proposed Problem 328.
Triangle, Incircle, Tangency Points, Parallel, Midpoint.

Tangential Quadrilateral

Tangential or Circumscribed Quadrilateral: Diagonals, Altitudes Theorem.

Right triangle Area, Semicircle, Square

Proposed Problem 327.
Right triangle Area, Incircle, Circumcircle, Square.

Equilateral triangle and semicircle

Proposed Problem 326.
Equilateral triangle, Semicircle, Equal arcs.

Tangential Quadrilateral

Tangential or Circumscribed Quadrilateral: Diagonals, Inradii Theorem.

Tangential or Circumscribed Quadrilateral Index, theorems and problems

Tangential or Circumscribed Quadrilateral
 

Six Tangential Quadrilaterals Theorem

Six Tangential or Circumscribed Quadrilaterals Theorem.

Tangential Quadrilateral: Pitot Theorem

Tangential or Circumscribed Quadrilateral: Pitot Theorem.

Isosceles triangle, Altitude, Incircle, Excircle

Proposed Problem 325.
Isosceles triangle, Altitude, Cevian, Incircle, Excircle.

Elearning 324: Quadrilateral

Proposed Problem 324.
Quadrilateral with perpendicular diagonals, Concurrence.

Altitude of a triangle

Altitude Index.
Theorems and Problems.

Triangle, Incenter, Circumcenter, Collinear

Proposed Problem 323.
Triangle, Incenter, Circumcenter, Equal circles, Collinearity.

Area of a triangle

Proposed Problem 322.
Square, Inscribed circle, Tangent, Triangle area.

Collinear Points Index

Collinear Points.

Incenter, Incircle, Index

Incenter. Index.

Triangle, Circumcenter, Incenter, Excenter, Collinear

Proposed Problem 320.
Triangle, Circumcircle, Incenter, Excenter, Collinear points.

Triangle, Altitude

Proposed Problem 319.
Triangle, Altitudes, Perpendiculars, Collinear points.

Elearning 318

Proposed Problem 318.
A Scalene triangle, Altitudes, Collinear points.

Elearning 317 Right triangle

Proposed Problem 317.
Right triangle and Inscribed Squares.

Circular Segments, Inscribed Squares

Proposed Problem 316.
Circular segments and Inscribed Squares.

Three Tangent Circles

Proposed Problem 315.
Three Tangent circles, Common external tangent line, Geometric Mean.

Polygons Index

Polygons.

Regular Octagon 

Proposed Problem 287: Regular Octagon.

Octagon Index

Octagons.

Octagon, Square

Proposed Problem 286: Square, Midpoints, Octagon.

Circula Sector, Semicircles, Circle

Proposed Problem 285: Circular Sector 90 degrees, Semicircles, Circle, Tangent, Radius.

Elearning 284: Circular Sector

Proposed Problem 284: Circular Sector 90 degrees, Semicircles, Tangent, Radius.

Circular Sector

Proposed Problem 283: Circular Sector 90 degrees, Semicircle, Circle inscribed, Radius.

Right triangle, Cevian, Perpendicular, Angles

Proposed Problem 282.
Right Triangle, Cevian, Angles, Perpendicular, Congruence.

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