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drawrect - iOS draw filled Circles

Not a graphics programmer here, so I'm trying to stumble through this. I'm trying to draw 9 filled circles, each a different color, each with a white border. The UIView's frame is CGRectMake (0,0,60,60). See attached image.

The problem is I'm getting "flat spots" on the borders on each side. Following is my code (from the UIView subclass):

- (void)drawRect:(CGRect)rect
{
    CGRect borderRect = CGRectMake(0.0, 0.0, 60.0, 60.0);
    CGContextRef context = UIGraphicsGetCurrentContext();
    CGContextSetRGBStrokeColor(context, 1.0, 1.0, 1.0, 1.0);
    CGContextSetRGBFillColor(context, colorRed, colorGreen, colorBlue, 1.0);
    CGContextSetLineWidth(context, 2.0);
    CGContextFillEllipseInRect (context, borderRect);
    CGContextStrokeEllipseInRect(context, borderRect);
    CGContextFillPath(context);
}

If I change to CGRectMake(0,0,56,56) in drawRect, I get flat spots only on the top and left sides, and the bottom & right sides look fine.

Can anyone suggest how I might fix this? It seems to me the border is being clipped by the UIView, but not knowing much about this, I really don't know how to fix it.

Thanks, in advance, for any of you graphics experts' suggestions.

enter image description here

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I like the answer from @AaronGolden, just wanted to add:

CGRect borderRect = CGRectInset(rect, 2, 2);

Or, better:

CGFloat lineWidth = 2;
CGRect borderRect = CGRectInset(rect, lineWidth * 0.5, lineWidth * 0.5);

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