There was a slight mistake in my code example: 
$ENV{HTTPS_PROXY} = 'http://proxy:8080';

[download]
Note the absence of trailing slash. It works for me, I've just got it running through my local squid proxy: 
[~/temp]$ ./test.pl
SSL_connect:before/connect initialization
SSL_connect:SSLv2/v3 write client hello A
SSL_connect:SSLv3 read server hello A
SSL_connect:SSLv3 read server certificate A
SSL_connect:SSLv3 read server done A
SSL_connect:SSLv3 write client key exchange A
SSL_connect:SSLv3 write change cipher spec A
SSL_connect:SSLv3 write finished A
SSL_connect:SSLv3 flush data
SSL_connect:SSLv3 read finished A
Successtext/htmlcharset=ISO-8859-1 document!

[download]

Please provide more information on your configuration. It works fine for me on Linux, running Perl 5.10.1 and LWP 5.835.

In reply to Re^3: LWP::Useragent GET fails with https by wazoox
in thread LWP::Useragent GET fails with https by Anonymous Monk

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