Maths WorkshopMaths problems with full solutions

Geometry problems

37 problems · Plane and space geometry, triangles, circles and analytic geometry.

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Ex.181Area from coordinates

Analytic geometry

Ex.324Distance from a point to a line

#lines #projection

Analytic geometry

Ex.339Thales' theorem

#circles #angles

Ex.609The point minimising the sum of distances to the vertices of a convex quadrilateral

#triangle inequality #extremal #convex quadrilateral

Ex.640Stewart's theorem and its consequences

#Stewart's theorem #medians #angle bisector #law of cosines

Ex.641Two classical locus problems

#locus #coordinates #perpendicular #circle with a given diameter

Analytic geometry

Ex.703Inradius of a right triangle

#incircle #right triangle

Ex.705The centroid of a triangle

#vectors #medians

Ex.1259The Gergonne point and Ceva's theorem

#Ceva's theorem #incircle #tangent lengths

Ex.1260Ptolemy and a point on the circumcircle of an equilateral triangle

#Ptolemy's theorem #cyclic quadrilateral #equilateral triangle

Ex.1261The area of a right triangle from the incircle touch point

#incircle #tangent lengths #Pythagoras #right triangle

Ex.1262The radical axis bisects the common tangent

#power of a point #radical axis #tangent

Ex.1263The reflection of the orthocentre lies on the circumcircle

#angle chasing #cyclic quadrilateral #orthocentre

Ex.1264Two circles, two secants and a pair of parallel lines

#directed angles #cyclic quadrilateral #parallel lines

Ex.1315Equilateral triangles and complex numbers

#complex numbers #equilateral triangle #roots of unity #rotation

Analytic geometry

Ex.1316Excircles and tangent lengths

#excircle #incircle #tangent lengths #semiperimeter #area

Ex.1317Pick's theorem and lattice triangles

#Pick's theorem #lattice points #area #equilateral triangle

Analytic geometry

Ex.1318The Apollonius circle

#locus #Apollonius circle #coordinates #angle bisector

Analytic geometry

Ex.1319The distance between the circumcentre and the orthocentre

#vectors #orthocentre #circumcentre #Euler line

Analytic geometry

Ex.1320Two circles and a line through a common point

#spiral similarity #two intersecting circles #chord length #projection

Ex.1321Weitzenböck's inequality

#geometric inequality #Heron's formula #area #sum of squares

Ex.1944The regular heptagon identity

#Ptolemy's theorem #regular heptagon #chords

Ex.1945Squares on two sides and the median

#complex numbers #squares #median #perpendicularity

Ex.1946The Euler line, by vectors

#Euler line #orthocentre #centroid #circumcentre #vectors

Ex.1947The three external bisectors meet the sides on a line

#Menelaus's theorem #angle bisector theorem #collinearity

Ex.1948Three equal products at the orthocentre

#radical axis #radical centre #orthocentre #circles on diameters

Ex.1949The one-seventh triangle

#area ratios #cevians #affine invariance #coordinates

Ex.1984Euler's formula OI² = R² − 2Rr

#Euler's formula #incentre #circumcircle #power of a point #incentre-excentre lemma

Ex.1985Harmonic division from Ceva and Menelaus

#Ceva #Menelaus #harmonic conjugates #cevians

Ex.1986Ptolemy's inequality by inversion

#inversion #Ptolemy's inequality #triangle inequality

Ex.1987The butterfly theorem

#butterfly theorem #chords #coordinates #Vieta

Analytic geometry

Ex.1988The symmedian and the tangents at B and C

#symmedian #tangent-chord angle #areas #law of sines

Ex.1989Trigonometric Ceva and a hidden isosceles triangle

#trigonometric Ceva #angle chasing #monotonicity

Ex.2228Napoleon's theorem

#complex numbers #rotations #equilateral triangle

Ex.2229The nine-point circle

#nine-point circle #complex numbers #orthocentre #feet of altitudes

Ex.2230The Simson line

#Simson line #complex numbers #collinearity #unit circle

Ex.2239The Erdős–Mordell inequality

#Erdős–Mordell #projections #pedal points #geometric inequality