幾何スタジオ
初級 → 数学オリンピック道具を集めて問題を解きましょう。好きな問題から始められます。持っていない道具が必要になったら、その場で学べます。
直線と三角形
11Six rays
≈ 450解説つきの例題Straight line
The outside angle
≈ 500Triangle sumStraight line
The bowtie
≈ 550Triangle sumVertical angles
Two equal sides
≈ 650Isosceles triangleTriangle sumStraight line
Between two parallels
≈ 650Parallel linesTriangle sum
The bent path
≈ 700Parallel lines
Bisector meets altitude
≈ 750Triangle sum
The wide bend
≈ 850Straight lineParallel lines
Three equal segments
≈ 1050Isosceles triangleExterior angleTriangle sum
The hidden circle
≈ 1100Isosceles triangleTriangle sum
The double bend
≈ 1100Parallel lines
多角形
2円
13On the diameter
≈ 600解説つきの例題Angle in a semicircleTriangle sum
Four points on a circle
≈ 700Cyclic quadrilateralStraight line
A diagonal in a cyclic quadrilateral
≈ 750Cyclic quadrilateral
Seen from the circle
≈ 800Inscribed angleTriangle sum
Where the tangent meets a chord
≈ 800Tangent and chordTriangle sum
Dropping the height
≈ 850Angle in a semicircleTriangle sum
Two chords cross
≈ 850Crossing chordsStraight line
A tangent at the apex
≈ 900Isosceles triangleTriangle sumTangent and chord
Two lines from outside
≈ 950Inscribed angleTwo secants
Two radii
≈ 1000Isosceles triangleTriangle sumInscribed angle
A tangent at T
≈ 1000Tangent meets radiusTriangle sumStraight lineIsosceles triangle
The hidden arc
≈ 1000Crossing chordsInscribed angle
Two tangents
≈ 1100Tangent meets radiusPolygon sumInscribed angle
長さと面積
15Across the square
≈ 700解説つきの例題Pythagoras
The equilateral triangle
≈ 750Pythagoras
A parallel cut
≈ 800Similar triangles
Two right triangles
≈ 850Pythagoras
Where the bisector lands
≈ 900Angle bisector theorem
Two crossing chords
≈ 900Power of a point
Joining the midpoints
≈ 950Midsegment
A tangent and a secant
≈ 1000Power of a point
Three similar triangles
≈ 1000Altitude on the hypotenusePythagoras
Scaling the area
≈ 1000Scale factor
The height nobody gave you
≈ 1050PythagorasArea two ways
A rhombus in a rectangle
≈ 1050Pythagoras
Stacking three coins
≈ 1150Special right triangles
Inside the hexagon
≈ 1150Polygon sumIsosceles triangleSpecial right triangles
The circle inside
≈ 1200PythagorasArea two waysInradius
影のついた部分の面積
20A coin in a box
≈ 500解説つきの例題Circle area
The corner that is left
≈ 500Circle area
Bite and bump
≈ 500Cut and slide
A half circle in a box
≈ 550Circle area
The ring
≈ 550Circle area
Three slices of eight
≈ 550Sector
The window
≈ 550Circle area
Four corners
≈ 600Circle area
Yin and yang
≈ 600Circle areaCut and slide
Two halves inside a half
≈ 650Circle area
Two squares, one overlap
≈ 650Overlap counted twice
A box in a coin
≈ 750PythagorasCircle area
The triangle in the half circle
≈ 800Angle in a semicirclePythagorasCircle areaArea two ways
The sliver
≈ 900SectorSpecial right triangles
Four petals
≈ 900Circle areaOverlap counted twice
Moons on a 6-8-10
≈ 1000Hippocrates’ moonsArea two ways
Hippocrates’ moons
≈ 1100PythagorasCircle areaArea two waysHippocrates’ moonsを獲得
A square under a dome
≈ 1150PythagorasCircle area
Two coins overlap
≈ 1300Triangle sumSectorSpecial right trianglesCut and slide
What’s left of the quarter
≈ 1500Circle areaSectorOverlap counted twiceCut and slide
動く図形
6対称移動と回転
4図形の構成
11Where the bisectors meet
≈ 950解説つきの例題Triangle sumIncentreを獲得
Where the altitudes meet
≈ 1000Polygon sumVertical anglesOrthocentreを獲得
Is there a circle?
≈ 1100Spotting a circleInscribed angle
The centre of the circle
≈ 1100Inscribed angleIsosceles triangle
A circle inside four sides
≈ 1100Equal tangents
A line from the top
≈ 1150Stewart's theorem
Two right angles, one circle
≈ 1250Spotting a circleCyclic quadrilateral
A point on the arc
≈ 1250Ptolemy
Tangents from the corners
≈ 1300Equal tangents
The pentagon's diagonal
≈ 1350Ptolemy
The triangle of feet
≈ 1450Spotting a circleCyclic quadrilateralStraight line
競技用の道具
10The wide angle
≈ 1200解説つきの例題Law of cosines
Three sides, one angle
≈ 1200Law of cosines
Two angles and a side
≈ 1200Law of sines
Weights on the corners
≈ 1300Mass points
Three lines through one point
≈ 1350Ceva's theorem
A line across the triangle
≈ 1400Menelaus' theorem
The circle through three corners
≈ 1400Law of cosinesLaw of sines
Equal powers
≈ 1450Radical axis
Two circles, one tangent
≈ 1500HomothetyPythagoras
How high is the centre?
≈ 1750PythagorasArea two waysLaw of sines
オリンピック
9Where the tangents meet
≈ 1350解説つきの例題Symmedian
The arc midpoint
≈ 1450Incentre–excentre lemma
Three centres on a line
≈ 1450Euler line
The nine-point circle
≈ 1500Euler lineNine-point circle
Turning the plane inside out
≈ 1500InversionSimilar triangles
Two bisectors, one line
≈ 1650Angle bisector theoremHarmonic division
A fourth circle
≈ 1800Cyclic quadrilateralSpotting a circleMiquel pointを獲得
Three feet on a line
≈ 1850Spotting a circleInscribed angleSimson lineを獲得
Down through the incentre
≈ 2000Angle bisector theoremStewart's theoremPower of a pointPtolemy