4.1 Geometry 209590.LetMbe a point in the plane of triangleABC. Prove that the centroids of the
trianglesMAB,MAC, andMCBform a triangle similar to triangleABC.
591.Find the locus of pointsPin the interior of a triangleABCsuch that the distances
fromPto the linesAB,BC, andCAare the side lengths of some triangle.
592.LetA 1 ,A 2 ,...,Anbe distinct points in the plane, and letmbe the number of
midpoints of all the segments they determine. What is the smallest value thatm
can have?
593.Given an acute-angled triangleABCwith altitudeAD, choose any pointMon
AD, and then drawBMand extend until it intersectsACinE, and drawCMand
extend until it intersectsABinF. Prove that∠ADE=∠ADF.
594.In a planar Cartesian system of coordinates consider a fixed pointP(a,b)and a
variable line throughP. LetAbe the intersection of the line with thex-axis.
ConnectAwith the midpointBof the segmentOP(Obeing the origin), and
throughC, which is the point of intersection of this line with they-axis, take the
parallel toOP. This parallel intersectsPAatM. Find the locus ofMas the line
varies.
595.LetABCDbe a parallelogram with unequal sides. LetEbe the foot of the per-
pendicular fromBtoAC. The perpendicular throughEtoBDintersectsBCinF
andABinG. Show thatEF=EGif and only ifABCDis a rectangle.
596.Find all pairs of real numbers(p, q) suchthat the inequality
∣∣
∣√
1 −x^2 −px−q∣∣
∣≤
√
2 − 1
2
holds for everyx∈[ 0 , 1 ].597.On the hyperbolaxy=1 consider four points whosex-coordinates arex 1 ,x 2 ,x 3 ,
andx 4. Show that if these points lie on a circle, thenx 1 x 2 x 3 x 4 =1.
The points of the plane can be represented as complex numbers. There are two
instances in which complex coordinates come in handy: in problems involving “nice’’
angles (such asπ 4 ,π 3 ,π 2 ), and in problems about regular polygons.
In complex coordinates the line passing through the pointsz 1 andz 2 has the parametric
equationz=tz 1 +( 1 −t)z 2 ,t∈R. Also, the angle between the line passing through
z 1 andz 2 and the line passing throughz 3 andz 4 is the argument of the complex number
z 1 −z 2
z 3 −z 4. The length of the segment determined by the pointsz^1 andz^2 is|z^1 −z^2 |. The
vertices of a regularn-gon can be chosen, up to a scaling factor, as 1,,^2 ,..., n−^1 ,
where=e^2 πi/n=cos^2 nπ+isin^2 nπ.